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Title: Encyclopaedia Britannica, 11th Edition, Volume 6, Slice 8 - "Conduction, Electric"
Author: Various
Language: English
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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
      printed in subscript. When letters are subscripted, they are
      preceded by an underscore, like C_n.

(2) Characters following a carat (^) were originally printed in
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(3) Letters topped by Macron are represented as [=x].

(4) [oo] stands for infinity; [int] for integral; [alpha], [beta], etc.
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(5) The following typographical errors have been corrected:

    Page 871: "an equation which is very useful, because it enables us,
     if we know the distribution." 'because' amended from 'becaus'.

    Page 890: "The colour of the luminosity due to positive rays is not
     in general the same as that due to anode rays." 'positive' amended
     from 'postive'.

    Page 890: "Anode Rays.--Gehrcke and Reichenhein (Ann. der Phys. 25,
     p. 861) have found that when the anode consists of a mixture of
     sodium and lithium chloride." 'mixture' amended from 'mixure'.



          ENCYCLOPAEDIA BRITANNICA

  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
           AND GENERAL INFORMATION

              ELEVENTH EDITION


           VOLUME VI, SLICE VIII

            Conduction, Electric



Article in This Slice:

  CONDUCTION, ELECTRIC



CONDUCTION, ELECTRIC. The electric conductivity of a substance is that
property in virtue of which all its parts come spontaneously to the same
electric potential if the substance is kept free from the operation of
electric force. Accordingly, the reciprocal quality, electric
resistivity, may be defined as a quality of a substance in virtue of
which a difference of potential can exist between different portions of
the body when these are in contact with some constant source of
electromotive force, in such a manner as to form part of an electric
circuit.

All material substances possess in some degree, large or small, electric
conductivity, and may for the sake of convenience be broadly divided
into five classes in this respect. Between these, however, there is no
sharply-marked dividing line, and the classification must therefore be
accepted as a more or less arbitrary one. These divisions are: (1)
metallic conductors, (2) non-metallic conductors, (3) dielectric
conductors, (4) electrolytic conductors, (5) gaseous conductors. The
first class comprises all metallic substances, and those mixtures or
combinations of metallic substances known as alloys. The second includes
such non-metallic bodies as carbon, silicon, many of the oxides and
peroxides of the metals, and probably also some oxides of the
non-metals, sulphides and selenides. Many of these substances, for
instance carbon and silicon, are well-known to have the property of
existing in several allotropic forms, and in some of these conditions,
so far from being fairly good conductors, they may be almost perfect
non-conductors. An example of this is seen in the case of carbon in its
three allotropic conditions--charcoal, graphite and diamond. As charcoal
it possesses a fairly well-marked but not very high conductivity in
comparison with metals; as graphite, a conductivity about
one-four-hundredth of that of iron; but as diamond so little
conductivity that the substance is included amongst insulators or
non-conductors. The third class includes those substances which are
generally called insulators or non-conductors, but which are better
denominated dielectric conductors; it comprises such solid substances as
mica, ebonite, shellac, india-rubber, gutta-percha, paraffin, and a
large number of liquids, chiefly hydrocarbons. These substances differ
greatly in insulating power, and according as the conductivity is more
or less marked, they are spoken of as bad or good insulators. Amongst
the latter many of the liquid gases hold a high position. Thus, liquid
oxygen and liquid air have been shown by Sir James Dewar to be almost
perfect non-conductors of electricity.

The behaviour of substances which fall into these three classes is
discussed below in section I., dealing with metallic conduction.

The fourth class, namely the electrolytic conductors comprises all those
substances which undergo chemical decomposition when they form part of
an electric circuit traversed by an electric current. They are discussed
in section II., dealing with electrolytic conduction.

The fifth and last class of conductors includes the gases. The
conditions under which this class of substance becomes possessed of
electric conductivity are considered in section III., on conduction in
gases.

In connexion with metallic conductors, it is a fact of great interest
and considerable practical importance, that, although the majority of
metals when in a finely divided or powdered condition are practically
non-conductors, a mass of metallic powder or filings may be made to pass
suddenly into a conductive condition by being exposed to the influence
of an electric wave. The same is true of the loose contact of two
metallic conductors. Thus if a steel point, such as a needle, presses
very lightly against a metallic plate, say of aluminium, it is found
that this metallic contact, if carefully adjusted, is non-conductive,
but that if an electric wave is created anywhere in the neighbourhood,
this non-conducting contact passes into a conductive state. This fact,
investigated and discovered independently by D. E. Hughes, C. Onesti, E.
Branly, O. J. Lodge and others, is applied in the construction of the
"coherer," or sensitive tube employed as a detector or receiver in that
form of "wireless telegraphy" chiefly developed by Marconi. Further
references to it are made in the articles ELECTRIC WAVES and TELEGRAPHY:
_Wireless_.

  _International Ohm._--The practical unit of electrical resistance was
  legally defined in Great Britain by the authority of the queen in
  council in 1894, as the "resistance offered to an invariable electric
  current by a column of mercury at the temperature of melting ice,
  14.4521 grammes in mass, of a constant cross-sectional area, and a
  length 106.3 centimetres." The same unit has been also legalized as a
  standard in France, Germany and the United States, and is denominated
  the "International or Standard Ohm." It is intended to represent as
  nearly as possible a resistance equal to 10° absolute C.G.S. units of
  electric resistance. Convenient multiples and subdivisions of the ohm
  are the microhm and the megohm, the former being a millionth part of
  an ohm, and the latter a million ohms. The resistivity of substances
  is then numerically expressed by stating the resistance of one cubic
  centimetre of the substance taken between opposed faces, and expressed
  in ohms, microhms or megohms, as may be most convenient. The
  reciprocal of the ohm is called the mho, which is the unit of
  conductivity, and is defined as the conductivity of a substance whose
  resistance is one ohm. The absolute unit of conductivity is the
  conductivity of a substance whose resistivity is one absolute C.G.S.
  unit, or one-thousandth-millionth part of an ohm. Resistivity is a
  quality in which material substances differ very widely. The metals
  and alloys, broadly speaking, are good conductors, and their
  resistivity is conveniently expressed in microhms per cubic
  centimetre, or in absolute C.G.S. units. Very small differences in
  density and in chemical purity make, however, immense differences in
  electric resistivity; hence the values given by different
  experimentalists for the resistivity of known metals differ to a
  considerable extent.


I. CONDUCTION IN SOLIDS

It is found convenient to express the resistivity of metals in two
different ways: (1) We may state the resistivity of one cubic centimetre
of the material in microhms or absolute units taken between opposed
faces. This is called the _volume-resistivity_; (2) we may express the
resistivity by stating the resistance in ohms offered by a wire of the
material in question of uniform cross-section one metre in length, and
one gramme in weight. This numerical measure of the resistivity is
called the _mass-resistivity_. The mass-resistivity of a body is
connected with its volume-resistivity and the density of the material in
the following manner:--The mass-resistivity, expressed in microhms per
metre-gramme, divided by 10 times the density is numerically equal to
the volume-resistivity per centimetre-cube in absolute C.G.S. units. The
mass-resistivity per metre-gramme can always be obtained by measuring
the resistance and the mass of any wire of uniform cross-section of
which the length is known, and if the density of the substance is then
measured, the volume-resistivity can be immediately calculated.

  If R is the resistance in ohms of a wire of length l, uniform
  cross-section s, and density d, then taking [rho] for the
  volume-resistivity we have 10^9R = [rho]l/s; but lsd = M, where M is
  the mass of the wire. Hence 10^9R = [rho]dl²/M. If l = 100 and M = 1,
  then R = [rho]'= resistivity in ohms per metre-gramme, and 10^9[rho]'
  = 10,000d[rho], or [rho] = 10^5[rho]'/d, and [rho]' = 10,000MR/l².

  The following rules, therefore, are useful in connexion with these
  measurements. To obtain the mass-resistivity per metre-gramme of a
  substance in the form of a uniform metallic wire:--Multiply together
  10,000 times the mass in grammes and the total resistance in ohms, and
  then divide by the square of the length in centimetres. Again, to
  obtain the volume-resistivity in C.G.S. units per centimetre-cube, the
  rule is to multiply the mass-resistivity in ohms by 100,000 and divide
  by the density. These rules, of course, apply only to wires of uniform
  cross-section. In the following Tables I., II. and III. are given the
  mass and volume resistivity of ordinary metals and certain alloys
  expressed in terms of the international ohm or the absolute C.G.S.
  unit of resistance, the values being calculated from the experiments
  of A. Matthiessen (1831-1870) between 1860 and 1865, and from later
  results obtained by J. A. Fleming and Sir James Dewar in 1893.

    TABLE I.--_Electric Mass-Resistivity of Various Metals at 0° C., or
    Resistance per Metre-gramme in International Ohms at 0° C._
    (Matthiessen.)

    +---------------------+---------------------+-------------+
    |                     | Resistance at 0° C. |             |
    |                     |  in International   | Approximate |
    |        Metal.       |   Ohms of a Wire    | Temperature |
    |                     |  1 Metre long and   | Coefficient |
    |                     |      Weighing       | near 20° C. |
    |                     |      1 Gramme.      |             |
    +---------------------+---------------------+-------------+
    |Silver (annealed)    |        .1523        |   0.00377   |
    |Silver (hard-drawn)  |        .1657        |      ..     |
    |Copper (annealed)    |        .1421        |   0.00388   |
    |Copper (hard-drawn)  | .1449 (Matthiessen's Standard)    |
    |Gold (annealed)      |        .4025        |   0.00365   |
    |Gold (hard-drawn)    |        .4094        |      ..     |
    |Aluminium (annealed) |        .0757        |      ..     |
    |Zinc (pressed)       |        .4013        |      ..     |
    |Platinum (annealed)  |       1.9337        |      ..     |
    |Iron (annealed)      |        .765         |      ..     |
    |Nickel (annealed)    |       1.058[1]      |      ..     |
    |Tin (pressed)        |        .9618        |   0.00365   |
    |Lead (pressed)       |       2.2268        |   0.00387   |
    |Antimony (pressed)   |       2.3787        |   0.00389   |
    |Bismuth (pressed)    |      12.8554[1]     |   0.00354   |
    |Mercury (liquid)     |      12.885[2]      |   0.00072   |
    +---------------------+---------------------+-------------+

  The data commonly used for calculating metallic resistivities were
  obtained by A. Matthiessen, and his results are set out in the Table
  II. which is taken from Cantor lectures given by Fleeming Jenkin in
  1866 at or about the date when the researches were made. The figures
  given by Jenkin have, however, been reduced to international ohms and
  C.G.S. units by multiplying by ([pi]/4) × 0.9866 × 10^5 = 77,485.

  Subsequently numerous determinations of the resistivity of various
  pure metals were made by Fleming and Dewar, whose results are set out
  in Table III.

    TABLE II.--_Electric Volume-Resistivity of Various Metals at 0° C.,
    or Resistance per Centimetre-cube in C.G.S. Units at 0° C._

    +---------------------+---------------------+
    |                     | Volume-Resistivity. |
    |       Metal.        | at 0° C. in C.G.S.  |
    |                     |        Units        |
    +---------------------+---------------------+
    |Silver (annealed)    |        1,502        |
    |Silver (hard-drawn)  |        1,629        |
    |Copper (annealed)    |        1,594        |
    |Copper (hard-drawn)  |        1,630[3]     |
    |Gold (annealed)      |        2,052        |
    |Gold (hard-drawn)    |        2,090        |
    |Aluminium (annealed) |        3,006        |
    |Zinc (pressed)       |        5,621        |
    |Platinum (annealed)  |        9,035        |
    |Iron (annealed)      |       10,568        |
    |Nickel (annealed)    |       12,429[4]     |
    |Tin (pressed)        |       13,178        |
    |Lead (pressed)       |       19,580        |
    |Antimony (pressed)   |       35,418        |
    |Bismuth (pressed)    |      130,872        |
    |Mercury (liquid)     |       94,896[5]     |
    +---------------------+---------------------+

    TABLE III.--_Electric Volume-Resistivity of Various Metals at 0° C.,
    or Resistance per Centimetre-cube at 0° C. in C.G.S. Units._
    (Fleming and Dewar, _Phil. Mag._, September 1893.)

    +--------------------------+------------------+------------------+
    |                          |    Resistance    | Mean Temperature |
    |          Metal.          |   at 0° C. per   |   Coefficient    |
    |                          | Centimetre-cube  |  between 0° C.   |
    |                          | in C.G.S. Units. |   and 100° C.    |
    +--------------------------+------------------+------------------+
    |Silver (electrolytic and  |                  |                  |
    |  well annealed)[6]       |      1,468       |     0.00400      |
    |Copper (electrolytic and  |                  |                  |
    |  well annealed)[6]       |      1,561       |     0.00428      |
    |Gold (annealed)           |      2,197       |     0.00377      |
    |Aluminium (annealed)      |      2,665       |     0.00435      |
    |Magnesium (pressed)       |      4,355       |     0.00381      |
    |Zinc                      |      5,751       |     0.00406      |
    |Nickel (electrolytic)[6]  |      6,935       |     0.00618      |
    |Iron (annealed)           |      9,065       |     0.00625      |
    |Cadmium                   |     10,023       |     0.00419      |
    |Palladium                 |     10,219       |     0.00354      |
    |Platinum (annealed)       |     10,917       |     0.003669     |
    |Tin (pressed)             |     13,048       |     0.00440      |
    |Thallium (pressed)        |     17,633       |     0.00398      |
    |Lead (pressed)            |     20,380       |     0.00411      |
    |Bismuth (electrolytic)[7] |    110,000       |     0.00433      |
    +--------------------------+------------------+------------------+

  _Resistivity of Mercury._--The volume-resistivity of pure mercury is a
  very important electric constant, and since 1880 many of the most
  competent experimentalists have directed their attention to the
  determination of its value. The experimental process has usually been
  to fill a glass tube of known dimensions, having large cup-like
  extensions at the ends, with pure mercury, and determine the absolute
  resistance of this column of metal. For the practical details of this
  method the following references may be consulted:--"The Specific
  Resistance of Mercury," Lord Rayleigh and Mrs Sidgwick, _Phil.
  Trans._, 1883, part i. p. 173, and R. T. Glazebrook, _Phil. Mag._,
  1885, p. 20; "On the Specific Resistance of Mercury," R. T. Glazebrook
  and T. C. Fitzpatrick, _Phil. Trans._, 1888, p. 179, or _Proc. Roy.
  Soc._, 1888, p. 44, or _Electrician_, 1888, 21, p. 538; "Recent
  Determinations of the Absolute Resistance of Mercury," R. T.
  Glazebrook, _Electrician_, 1890, 25, pp. 543 and 588. Also see J. V.
  Jones, "On the Determination of the Specific Resistance of Mercury in
  Absolute Measure," _Phil. Trans._, 1891, A, p. 2. Table IV. gives the
  values of the volume-resistivity of mercury as determined by various
  observers, the constant being expressed (a) in terms of the resistance
  in ohms of a column of mercury one millimetre in cross-section and 100
  centimetres in length, taken at 0° C.; and (b) in terms of the length
  in centimetres of a column of mercury one square millimetre in
  cross-section taken at 0° C. The result of all the most careful
  determinations has been to show that the resistivity of pure mercury
  at 0° C. is about 94,070 C.G.S. electromagnetic units of resistance,
  and that a column of mercury 106.3 centimetres in length having a
  cross-sectional area of one square millimetre would have a resistance
  at 0° C. of one international ohm. These values have accordingly been
  accepted as the official and recognized values for the specific
  resistance of mercury, and the definition of the ohm. The table also
  states the methods which have been adopted by the different observers
  for obtaining the absolute value of the resistance of a known column
  of mercury, or of a resistance coil afterwards compared with a known
  column of mercury. A column of figures is added showing the value in
  fractions of an international ohm of the British Association Unit
  (B.A.U.), formerly supposed to represent the true ohm. The real value
  of the B.A.U. is now taken as .9866 of an international ohm.

    TABLE IV.--_Determinations of the Absolute Value of the
    Volume-Resistivity of Mercury and the Mercury Equivalent of the
    Ohm._

    +-----------------+------+----------------------+--------+----------+---------+
    |                 |      |                      |        |Value of  |Value of |
    |                 |      |                      |Value of|100 Centi-|Ohm in   |
    |    Observer.    | Date.|        Method.       |B.A.U.  |metres of |Centi-   |
    |                 |      |                      |in Ohms.|Mercury   |metres of|
    |                 |      |                      |        |in Ohms.  |Mercury. |
    +-----------------+------+----------------------+--------+----------+---------+
    |Lord Rayleigh    | 1882 | Rotating coil        | .98651 |  .94133  | 106.31  |
    |Lord Rayleigh    | 1883 | Lorenz method        | .98677 |    ..    | 106.27  |
    |G. Wiedemann     | 1884 | Rotation through 180°|   ..   |    ..    | 106.19  |
    |E. E. N. Mascart | 1884 | Induced current      | .98611 |  .94096  | 106.33  |
    |H. A. Rowland    | 1887 | Mean of several      | .98644 |  .94071  | 106.32  |
    |                 |      |   methods            |        |          |         |
    |F. Kohlrausch    | 1887 | Damping of magnets   | .98660 |  .94061  | 106.32  |
    |R. T. Glazebrook | 1882 | Induced currents     | .98665 |  .94074  | 106.29  |
    |                 | 1888 |                      |        |          |         |
    |Wuilleumeier     | 1890 |                      | .98686 |  .94077  | 106.31  |
    |Duncan and Wilkes| 1890 | Lorenz               | .98634 |  .94067  | 106.34  |
    |J. V. Jones      | 1891 | Lorenz               |   ..   |  .94067  | 106.31  |
    |                 |      |                      +--------+----------+---------+
    |                 |      | Mean value             .98653 |          |         |
    |                 |      |                               |          |         |
    |Streker          | 1885 | An absolute determin-|        |  .94056  | 106.32  |
    |Hutchinson       | 1888 |   ation of resistance|        |  .94074  | 106.30  |
    |E. Salvioni      | 1890 |   was not made. The  |        |  .94054  | 106.33  |
    |E. Salvioni      |  ..  |   value .98656 has   |        |  .94076  | 106.30  |
    |                 |      |   been used          |        |          |         |
    |                 |      |                      |        +----------+---------+
    |                 |      |                    Mean value    .94076  | 106.31  |
    |                 |      |                                          |         |
    |H. F. Weber      | 1884 | Induced current      | Absolute measure- | 105.37  |
    |H. F. Weber      |  ..  | Rotating coil        | ments compared    | 106.16  |
    |A. Roiti         | 1884 | Mean effect of       | with German silver| 105.89  |
    |                 |      |   induced current    | wire coils issued |         |
    |F. Himstedt      | 1885 |                      | by Siemens and    | 105.98  |
    |                 |      |                      | Streker           |         |
    |F. E. Dorn       | 1889 | Damping of a magnet  |                   | 106.24  |
    |Wild             | 1883 | Damping of a magnet  |                   | 106.03  |
    |L. V. Lorenz     | 1885 | Lorenz method        |                   | 105.93  |
    +-----------------+------+----------------------+-------------------+---------+

  For a critical discussion of the methods which have been adopted in
  the absolute determination of the resistivity of mercury, and the
  value of the British Association unit of resistance, the reader may be
  referred to the _British Association Reports_ for 1890 and 1892
  (_Report of Electrical Standards Committee_), and to the
  _Electrician_, 25, p. 456, and 29, p. 462. A discussion of the
  relative value of the results obtained between 1882 and 1890 was given
  by R. T. Glazebrook in a paper presented to the British Association at
  Leeds, 1890.

  _Resistivity of Copper._--In connexion with electro-technical work the
  determination of the conductivity or resistivity values of annealed
  and hard-drawn copper wire at standard temperatures is a very
  important matter. Matthiessen devoted considerable attention to this
  subject between the years 1860 and 1864 (see _Phil. Trans._, 1860, p.
  150), and since that time much additional work has been carried out.
  Matthiessen's value, known as _Matthiessen's Standard_, for the
  mass-resistivity of pure hard-drawn copper wire, is the resistance of
  a wire of pure hard-drawn copper one metre long and weighing one
  gramme, and this is equal to 0.14493 international ohms at 0° C. For
  many purposes it is more convenient to express temperature in
  Fahrenheit degrees, and the recommendation of the 1899 committee on
  copper conductors[8] is as follows:--"Matthiessen's standard for
  hard-drawn conductivity commercial copper shall be considered to be
  the resistance of a wire of pure hard-drawn copper one metre long,
  weighing one gramme which at 60° F. is 0.153858 international ohms."
  Matthiessen also measured the mass-resistivity of annealed copper, and
  found that its conductivity is greater than that of hard-drawn copper
  by about 2.25% to 2.5% As annealed copper may vary considerably in its
  state of annealing, and is always somewhat hardened by bending and
  winding, it is found in practice that the resistivity of commercial
  annealed copper is about 1¼% less than that of hard-drawn copper. The
  standard now accepted for such copper, on the recommendation of the
  1899 Committee, is a wire of pure annealed copper one metre long,
  weighing one gramme, whose resistance at 0° C. is 0.1421 international
  ohms, or at 60° F., 0.150822 international ohms. The specific gravity
  of copper varies from about 8.89 to 8.95, and the standard value
  accepted for high conductivity commercial copper is 8.912,
  corresponding to a weight of 555 lb per cubic foot at 60° F. Hence the
  volume-resistivity of pure annealed copper at 0° C. is 1.594 microhms
  per c.c., or 1594 C.G.S. units, and that of pure hard-drawn copper at
  0° C. is 1.626 microhms per c.c., or 1626 C.G.S. units. Since
  Matthiessen's researches, the most careful scientific investigation on
  the conductivity of copper is that of T. C. Fitzpatrick, carried out
  in 1890. (_Brit. Assoc. Report_, 1890, Appendix 3, p. 120.)
  Fitzpatrick confirmed Matthiessen's chief result, and obtained values
  for the resistivity of hard-drawn copper which, when corrected for
  temperature variation, are in entire agreement with those of
  Matthiessen at the same temperature.

The volume resistivity of alloys is, generally speaking, much higher
than that of pure metals. Table V. shows the volume resistivity at 0° C.
of a number of well-known alloys, with their chemical composition.

    TABLE V.--_Volume-Resistivity of Alloys of known Composition at 0°
    C. in C.G.S. Units per Centimetre-cube. Mean Temperature
    Coefficients taken at 15° C._ (Fleming and Dewar.)

    +-----------------------+-----------+-----------+-------------------+
    |                       |Resistivity|Temperature|Composition in per |
    |        Alloys.        |at 0° C.   |Coefficient|      cents.       |
    |                       |           |at 15° C.  |                   |
    +-----------------------+-----------+-----------+-------------------+
    |Platinum-silver        |  31,582   |  .000243  |Pt 33%, Ag 66%     |
    |Platinum-iridium       |  30,896   |  .000822  |Pt 80%, Ir 20%     |
    |Platinum-rhodium       |  21,142   |  .00143   |Pt 90%, Rd 10%     |
    |Gold-silver            |   6,280   |  .00124   |Au 90%, Ag 10%     |
    |Manganese-steel        |  67,148   |  .00127   |Mn 12%, Fe 78%     |
    |Nickel-steel           |  29,452   |  .00201   |Ni 4.35%, remaining|
    |                       |           |           |  percentage       |
    |                       |           |           |  chiefly iron, but|
    |                       |           |           |  uncertain        |
    |German silver          |  29,982   |  .000273  |Cu5Zn3Ni2          |
    |Platinoid[9]           |  41,731   |  .00031   |                   |
    |Manganin               |  46,678   |  .0000    |Cu 84%, Mn 12%,    |
    |                       |           |           |Ni 4%              |
    |Aluminium-silver       |   4,641   |  .00238   |Al 94%, Ag 6%      |
    |Aluminium-copper       |   2,904   |  .00381   |Al 94%, Cu 6%      |
    |Copper-aluminium       |   8,847   |  .000897  |Cu 97%, Al 3%      |
    |Copper-nickel-aluminium|  14,912   |  .000643  |Cu 87%, Ni 6.5%,   |
    |                       |           |           |  Al 6.5%          |
    |Titanium-aluminium     |   3,887   |  .00290   |                   |
    +-----------------------+-----------+-----------+-------------------+

  Generally speaking, an alloy having high resistivity has poor
  mechanical qualities, that is to say, its tensile strength and
  ductility are small. It is possible to form alloys having a
  resistivity as high as 100 microhms per cubic centimetre; but, on the
  other hand, the value of an alloy for electro-technical purposes is
  judged not merely by its resistivity, but also by the degree to which
  its resistivity varies with temperature, and by its capability of
  being easily drawn into fine wire of not very small tensile strength.
  Some pure metals when alloyed with a small proportion of another metal
  do not suffer much change in resistivity, but in other cases the
  resultant alloy has a much higher resistivity. Thus an alloy of pure
  copper with 3% of aluminium has a resistivity about 5½ times that of
  copper; but if pure aluminium is alloyed with 6% of copper, the
  resistivity of the product is not more than 20% greater than that of
  pure aluminium. The presence of a very small proportion of a
  non-metallic element in a metallic mass, such as oxygen, sulphur or
  phosphorus, has a very great effect in increasing the resistivity.
  Certain metallic elements also have the same power; thus platinoid has
  a resistivity 30% greater than German silver, though it differs from
  it merely in containing a trace of tungsten.

The resistivity of non-metallic conductors is in all cases higher than
that of any pure metal. The resistivity of carbon, for instance, in the
forms of charcoal or carbonized organic material and graphite, varies
from 600 to 6000 microhms per cubic centimetre, as shown in Table VI.:--

  TABLE VI.--_Electric Volume-Resistivity in Microhms per
  Centimetre-cube of Various Forms of Carbon at 15° C._

  +-----------------------------------------+---------------+
  |                Substance.               |  Resistivity. |
  +-----------------------------------------+---------------+
  | Arc lamp carbon rod                     |  8000         |
  | Jablochkoff candle carbon               |  4000         |
  | Carré carbon                            |  3400         |
  | Carbonized bamboo                       |  6000         |
  | Carbonized parchmentized thread         |  4000 to 5000 |
  | Ordinary carbon filament from glow-lamp |               |
  |   "treated" or flashed                  |  2400 to 2500 |
  | Deposited or secondary carbon           |   600 to  900 |
  | Graphite                                |   400 to  500 |
  +-----------------------------------------+---------------+

The resistivity of liquids is, generally speaking, much higher than that
of any metals, metallic alloys or non-metallic conductors. Thus fused
lead chloride, one of the best conducting liquids, has a resistivity in
its fused condition of 0.376 ohm per centimetre-cube, or 376,000
microhms per centimetre-cube, whereas that of metallic alloys only in
few cases exceeds 100 microhms per centimetre-cube. The resistivity of
solutions of metallic salts also varies very largely with the proportion
of the diluent or solvent, and in some instances, as in the aqueous
solutions of mineral acids; there is a maximum conductivity
corresponding to a certain dilution. The resistivity of many liquids,
such as alcohol, ether, benzene and pure water, is so high, in other
words, their conductivity is so small, that they are practically
insulators, and the resistivity can only be appropriately expressed in
megohms per centimetre-cube.

In Table VII. are given the names of a few of these badly-conducting
liquids, with the values of their volume-resistivity in megohms per
centimetre-cube:--

  TABLE VII.--_Electric Volume-Resistivity of Various Badly-Conducting
  Liquids in Megohms per Centimetre-cube._

  +---------------------------+----------------+---------------------+
  |                           |  Resistivity   |                     |
  |         Substance.        |  in Megohms    |      Observer.      |
  |                           |   per c.c.     |                     |
  +---------------------------+----------------+---------------------+
  | Ethyl alcohol             |     0.5        | Pfeiffer.           |
  | Ethyl ether               | 1.175 to 3.760 | W. Kohlrausch.      |
  | Benzene                   |    4.700       |                     |
  | Absolutely pure water     | 25.0 at 18° C. | Value estimated     |
  |  approximates probably to |                |  by F. Kohlrausch   |
  |                           |                |  and A. Heydweiler. |
  | All very dilute aqueous   | 1.00 at 18° C. | From results by     |
  |  salt solutions having a  |                |  F. Kohlrausch      |
  |  concentration of about   |                |  and others.        |
  |  0.00001 of an equivalent |                |                     |
  |  gramme molecule[10] per  |                |                     |
  |  litre approximate to     |                |                     |
  +---------------------------+----------------+---------------------+

The resistivity of all those substances which are generally called
dielectrics or insulators is also so high that it can only be
appropriately expressed in millions of megohms per centimetre-cube, or
in megohms per quadrant-cube, the quadrant being a cube the side of
which is 10^9 cms. (see Table VIII.).

Effects of Heat.--Temperature affects the resistivity of these different
classes of conductors in different ways. In all cases, so far as is yet
known, the resistivity of a pure metal is increased if its temperature is
raised, and decreased if the temperature is lowered, so that if it could
be brought to the absolute zero of temperature (-273° C.) its resistivity
would be reduced to a very small fraction of its resistance at ordinary
temperatures. With metallic alloys, however, rise of temperature does not
always increase resistivity: it sometimes diminishes it, so that many
alloys are known which have a maximum resistivity corresponding to a
certain temperature, and at or near this point they vary very little in
resistance with temperature. Such alloys have, therefore, a negative
temperature-variation of resistance at and above fixed temperatures.
Prominent amongst these metallic compounds are alloys of iron, manganese,
nickel and copper, some of which were discovered by Edward Weston, in the
United States. One well-known alloy of copper, manganese and nickel, now
called manganin, which was brought to the notice of electricians by the
careful investigations made at the Berlin Physikalisch-Technische
Reichsanstalt, is characterized by having a zero temperature coefficient
at or about a certain temperature in the neighbourhood of 15° C. Hence
within a certain range of temperature on either side of this critical
value the resistivity of manganin is hardly affected at all by
temperature. Similar alloys can be produced from copper and
ferro-manganese. An alloy formed of 80% copper and 20% manganese in an
annealed condition has a nearly zero temperature-variation of resistance
between 20° C. and 100° C. In the case of non-metals the action of
temperature is generally to diminish the resistivity as temperature
rises, though this is not universally so. The interesting observation has
been recorded by J. W. Howell, that "treated" carbon filaments and
graphite are substances which have a minimum resistance corresponding to
a certain temperature approaching red heat (_Electrician_, vol. xxxviii.
p. 835). At and beyond this temperature increased heating appears to
increase their resistivity; this phenomenon may, however, be accompanied
by a molecular change and not be a true temperature variation. In the
case of dielectric conductors and of electrolytes, the action of rising
temperature is to reduce resistivity. Many of the so-called insulators,
such as mica, ebonite, indiarubber, and the insulating oils, paraffin,
&c., decrease in resistivity with great rapidity as the temperature
rises. With guttapercha a rise in temperature from 0° C. to 24° C. is
sufficient to reduce the resistivity of one-twentieth part of its value
at 0° C., and the resistivity of flint glass at 140° C. is only
one-hundredth of what it is at 60° C.

  TABLE VIII.--_Electric Volume-Resistivity of Dielectrics reckoned in
  Millions of Megohms (Mega-megohms) per Centimetre-cube, and in Megohms
  per Quadrant-cube, i.e. a Cube whose Side is 10^9 cms._

  +------------------------+---------------------------+----------+
  |                        |         Resistivity.      |          |
  |                        +------------+--------------+ Tempera- |
  |       Substance.       |Mega-megohms| Megohms per  |   tura   |
  |                        | per c.c.   |Quadrant-cube.|   Cent.  |
  +------------------------+------------+--------------+----------+
  |Bohemian glass          |       61   |      .061    |    60°   |
  |Mica                    |       84   |      .084    |    20°   |
  |Gutta-percha            |      450   |      .45     |    24°   |
  |Flint glass             |    1,020   |     1.02     |    60°   |
  |Glover's vulcanized     |            |              |          |
  | indiarubber            |    1,630   |     1.63     |    15°   |
  |Siemens' ordinary pure  |            |              |          |
  | vulcanized indiarubber |    2,280   |     2.28     |    15°   |
  |Shellac                 |    9,000   |     9.0      |    28°   |
  |Indiarubber             |   10,900   |    10.9      |    24°   |
  |Siemens' high-insulating|            |              |          |
  | fibrous material       |   11,900   |    11.9      |    15°   |
  |Siemens' special        |            |              |          |
  | high-insulating        |            |              |          |
  | indiarubber.           |   16,170   |    16.17     |    15°   |
  |Flint glass             |   20,000   |    20.0      |    20°   |
  |Ebonite                 |   28,000   |    28.       |    46°   |
  |Paraffin                |   34,000   |    34.       |    46°   |
  +------------------------+------------+--------------+----------+

  A definition may here be given of the meaning of the term _Temperature
  Coefficient_. If, in the first place, we suppose that the resistivity
  ([rho]t) at any temperature (t) is a simple linear function of the
  resistivity ([rho]0) at 0° C., then we can write [rho]t = [rho]0(1 +
  [alpha]t), or [alpha] = ([rho]t - [rho]0)/[rho]0t.

  The quantity [alpha] is then called the temperature-coefficient, and
  its reciprocal is the temperature at which the resistivity would
  become zero. By an extension of this notion we can call the quantity
  d[rho]/[rho]dt the temperature coefficient corresponding to any
  temperature t at which the resistivity is [rho]. In all cases the
  relation between the resistivity of a substance and the temperature is
  best set out in the form of a curve called a temperature-resistance
  curve. If a series of such curves are drawn for various pure metals,
  temperature being taken as abscissa and resistance as ordinate, and if
  the temperature range extends from the absolute zero of temperature
  upwards, then it is found that these temperature-resistance lines are
  curved lines having their convexity either upwards or downwards. In
  other words, the second differential coefficient of resistance with
  respect to temperature is either a positive or negative quantity. An
  extensive series of observations concerning the form of the
  resistivity curves for various pure metals over a range of temperature
  extending from -200° C. to +200° C. was carried out in 1892 and 1893
  by Fleming and Dewar (_Phil. Mag._ Oct. 1892 and Sept. 1893). The
  resistance observations were taken with resistance coils constructed
  with wires of various metals obtained in a state of great chemical
  purity. The lengths and mean diameters of the wires were carefully
  measured, and their resistance was then taken at certain known
  temperatures obtained by immersing the coils in boiling aniline,
  boiling water, melting ice, melting carbonic acid in ether, and
  boiling liquid oxygen, the temperatures thus given being +184°.5 C.,
  +100° C., 0° C., -78°.2 C. and -182°.5 C. The resistivities of the
  various metals were then calculated and set out in terms of the
  temperature. From these data a chart was prepared showing the
  temperature-resistance curves of these metals throughout a range of
  400 degrees. The exact form of these curves through the region of
  temperature lying between -200° C. and -273° C. is not yet known. As
  shown on the chart, the curves evidently do not converge to precisely
  the same point. It is, however, much less probable that the resistance
  of any metal should vanish at a temperature above the absolute zero
  than at the absolute zero itself, and the precise path of these curves
  at their lower ends cannot be delineated until means are found for
  fixing independently the temperature of some regions in which the
  resistance of metallic wires can be measured. Sir J. Dewar
  subsequently showed that for certain pure metals it is clear that the
  resistance would not vanish at the absolute zero but would be reduced
  to a finite but small value (see "Electric Resistance Thermometry at
  the Temperature of Boiling Hydrogen," _Proc. Roy. Soc._ 1904, 73, p.
  244).

  The resistivity curves of the magnetic metals are also remarkable for
  the change of curvature they exhibit at the magnetic critical
  temperature. Thus J. Hopkinson and D. K. Morris (_Phil. Mag._
  September 1897, p. 213) observed the remarkable alteration that takes
  place in the iron resistance temperature curve in the neighbourhood of
  780° C. At that temperature the direction of the curvature of the
  curve changes so that it becomes convex upwards instead of convex
  downwards, and in addition the value of the temperature coefficient
  undergoes a great reduction. The mean temperature coefficient of iron
  in the neighbourhood of 0° C. is 0.0057; at 765° C. it rises to a
  maximum value 0.0204; but at 1000° C. it falls again to a lower value,
  0.00244. A similar rise to a maximum value and subsequent fall are
  also noted in the case of the specific heat of iron. The changes in
  the curvature of the resistivity curves are undoubtedly connected with
  the molecular changes that occur in the magnetic metals at their
  critical temperatures.

  A fact of considerable interest in connexion with resistivity is the
  influence exerted by a strong magnetic field in the case of some
  metals, notably bismuth. It was discovered by A. Righi and confirmed
  by S. A. Leduc (_Journ. de Phys._ 1886, 5, p. 116, and 1887, 6, p.
  189) that if a pure bismuth wire is placed in a magnetic field
  transversely to the direction of the magnetic field, its resistance is
  considerably increased. This increase is greatly affected by the
  temperature of the metal (Dewar and Fleming, _Proc. Roy. Soc._ 1897,
  60, p. 427). The temperature coefficient of pure copper is an
  important constant, and its value as determined by Messrs Clark, Forde
  and Taylor in terms of Fahrenheit temperature is

  [rho]t = [rho]32 {1 + 0.0023708(t - 32) + 0.0000034548(t - 32)²}.

_Time Effects._--In the case of dielectric conductors, commonly called
insulators, such as indiarubber, guttapercha, glass and mica, the
electric resistivity is not only a function of the temperature but also
of the time during which the electromotive force employed to measure it
is imposed. Thus if an indiarubber-covered cable is immersed in water
and the resistance of the dielectric between the copper conductor and
the water measured by ascertaining the current which can be caused to
flow through it by an electromotive force, this current is found to vary
very rapidly with the time during which the electromotive force is
applied. Apart from the small initial effect due to the electrostatic
capacity of the cable, the application of an electromotive force to the
dielectric produces a current through it which rapidly falls in value,
as if the electric resistance of the dielectric were increasing. The
current, however, does not fall continuously but tends to a limiting
value, and it appears that if the electromotive force is kept applied
to the cable for a prolonged time, a small and nearly constant current
will ultimately be found flowing through it. It is customary in
electro-technical work to consider the resistivity of the dielectric as
the value it has after the electromotive force has been applied for one
minute, the standard temperature being 75° F. This, however, is a purely
conventional proceeding, and the number so obtained does not necessarily
represent the true or ohmic resistance of the dielectric. If the
electromotive force is increased, in the case of a large number of
ordinary dielectrics the apparent resistance at the end of one minute's
electrification decreases as the electromotive force increases.

_Practical Standards._--The practical measurement of resistivity
involves many processes and instruments (see WHEATSTONE'S BRIDGE and
OHMMETER). Broadly speaking, the processes are divided into _Comparison
Methods_ and _Absolute Methods_. In the former a comparison is effected
between the resistance of a material in a known form and some standard
resistance. In the _Absolute Methods_ the resistivity is determined
without reference to any other substance, but with reference only to the
fundamental standards of length, mass and time. Immense labour has been
expended in investigations concerned with the production of a standard
of resistance and its evaluation in absolute measure. In some cases the
absolute standard is constructed by filling a carefully-calibrated tube
of glass with mercury, in order to realize in a material form the
official definition of the ohm; in this manner most of the principal
national physical laboratories have been provided with standard mercury
ohms. (For a full description of the standard mercury ohm of the Berlin
Physikalisch-Technische Reichsanstalt, see the _Electrician_, xxxvii.
569.) For practical purposes it is more convenient to employ a standard
of resistance made of wire.

  Opinion is not yet perfectly settled on the question whether a wire
  made of any alloy can be considered to be a perfectly unalterable
  standard of resistance, but experience has shown that a platinum
  silver alloy (66% silver, 33% platinum), and also the alloy called
  manganin, seem to possess the qualities of permanence essential for a
  wire-resistance standard. A comparison made in 1892 and 1894 of all
  the manganin wire copies of the ohm made at the Reichsanstalt in
  Berlin, showed that these standards had remained constant for two
  years to within one or two parts in 100,000. It appears, however, that
  in order that manganin may remain constant in resistivity when used in
  the manufacture of a resistance coil, it is necessary that the alloy
  should be _aged_ by heating it to a temperature of 140° C. for ten
  hours; and to prevent subsequent changes in resistivity, solders
  containing zinc must be avoided, and a silver solder containing 75% of
  silver employed in soldering the manganin wire to its connexions.

The authorities of the Berlin Reichsanstalt have devoted considerable
attention to the question of the best form for a wire standard of
electric resistance. In that now adopted the resistance wire is
carefully insulated and wound on a brass cylinder, being doubled on
itself to annul inductance as much as possible. In the coil two wires
are wound on in parallel, one being much finer than the other, and the
final adjustment of the coil to an exact value is made by shortening the
finer of the two. A standard of resistance for use in a laboratory now
generally consists of a wire of manganin or platinum-silver carefully
insulated and enclosed in a brass case. Thick copper rods are connected
to the terminals of the wire in the interior of the case, and brought to
the outside, being carefully insulated at the same time from one another
and from the case. The coil so constructed can be placed under water or
paraffin oil, the temperature of which can be exactly observed during
the process of taking a resistance measurement. Equalization of the
temperature of the surrounding medium is effected by the employment of a
stirrer, worked by hand or by a small electric motor. The construction
of a standard of electrical resistance consisting of mercury in a glass
tube is an operation requiring considerable precautions, and only to be
undertaken by those experienced in the matter. Opinions are divided on
the question whether greater permanence in resistance can be secured by
mercury-in-glass standards of resistance or by wire standards, but the
latter are at least more portable and less fragile.

  A full description of the construction of a standard wire-resistance
  coil on the plan adopted by the Berlin Physikalisch-Technische
  Reichsanstalt is given in the Report of the British Association
  Committee on Electrical Standards, presented at the Edinburgh Meeting
  in 1892. For the design and construction of standards of electric
  resistances adapted for employment in the comparison and measurement
  of very low or very high resistances, the reader may be referred to
  standard treatises on electric measurements.

  BIBLIOGRAPHY.--See also J. A. Fleming, _A Handbook for the Electrical
  Laboratory and Testing Room_, vol. i. (London, 1901); _Reports of the
  British Association Committee on Electrical Standards_, edited by
  Fleeming Jenkin (London, 1873); A. Matthiessen and C. Vogt, "On the
  Influence of Temperature on the Conducting Power of Alloys," _Phil.
  Trans._, 1864, 154, p. 167, and _Phil. Mag._, 1865, 29, p. 363; A.
  Matthiessen and M. Holtzmann, "On the Effect of the Presence of Metals
  and Metalloids upon the Electric Conducting Power of Pure Copper,"
  _Phil. Trans._, 1860, 150, p. 85; T. C. Fitzpatrick, "On the Specific
  Resistance of Copper," _Brit. Assoc. Report_, 1890, p. 120, or
  _Electrician_, 1890, 25, p. 608; R. Appleyard, _The Conductometer and
  Electrical Conductivity_; Clark, Forde and Taylor, _Temperature
  Coefficients of Copper_ (London, 1901).     (J. A. F.)


II. CONDUCTION IN LIQUIDS

Through liquid metals, such as mercury at ordinary temperatures and
other metals at temperatures above their melting points, the electric
current flows as in solid metals without changing the state of the
conductor, except in so far as heat is developed by the electric
resistance. But another class of liquid conductors exists, and in them
the phenomena are quite different. The conductivity of fused salts, and
of solutions of salts and acids, although less than that of metals, is
very great compared with the traces of conductivity found in so-called
non-conductors. In fused salts and conducting solutions the passage of
the current is always accompanied by definite chemical changes; the
substance of the conductor or electrolyte is decomposed, and the
products of the decomposition appear at the electrodes, i.e. the
metallic plates by means of which the current is led into and out of the
solution. The chemical phenomena are considered in the article
ELECTROLYSIS; we are here concerned solely with the mechanism of this
_electrolytic_ conduction of the current.

To explain the appearance of the products of decomposition at the
electrodes only, while the intervening solution is unaltered, we suppose
that, under the action of the electric forces, the opposite parts of the
electrolyte move in opposite directions through the liquid. These
opposite parts, named ions by Faraday, must therefore be associated with
electric charges, and it is the convective movement of the opposite
streams of ions carrying their charges with them that, on this view,
constitutes the electric current.

In metallic conduction it is found that the current is proportional to
the applied electromotive force--a relation known by the name of Ohm's
law. If we place in a circuit with a small electromotive force an
electrolytic cell consisting of two platinum electrodes and a solution,
the initial current soon dies away, and we shall find that a certain
minimum electromotive force must be applied to the circuit before any
considerable permanent current passes. The chemical changes which are
initiated on the surfaces of the electrodes set up a reverse
electromotive force of polarization, and, until this is overcome, only a
minute current, probably due to the slow but steady removal of the
products of decomposition from the electrodes by a process of diffusion,
will pass through the cell. Thus it is evident that, considering the
electrolytic cell as a whole, the passage of the current through it
cannot conform to Ohm's law. But the polarization is due to chemical
changes, which are confined to the surfaces of the electrodes; and it is
necessary to inquire whether, if the polarization at the electrodes be
eliminated, the passage of the current through the bulk of the solution
itself is proportional to the electromotive force actually applied to
that solution. Rough experiment shows that the current is proportional
to the excess of the electromotive force over a constant value, and thus
verifies the law approximately, the constant electromotive force to be
overcome being a measure of the polarization. A more satisfactory
examination of the question was made by F. Kohlrausch in the years 1873
to 1876. Ohm's law states that the current C is proportional to the
electromotive force E, or C = kR, where k is a constant called the
conductivity of the circuit. The equation may also be written as C =
E/R, where R is a constant, the reciprocal of k, known as the resistance
of the circuit. The essence of the law is the proportionality between C
and E, which means that the ratio E/C is a constant. But E/C = R, and
thus the law may be tested by examining the constancy of the measured
resistance of a conductor when different currents are passing through
it. In this way Ohm's law has been confirmed in the case of metallic
conduction to a very high degree of accuracy. A similar principle was
applied by Kohlrausch to the case of electrolytes, and he was the first
to show that an electrolyte possesses a definite resistance which has a
constant value when measured with different currents and by different
experimental methods.

_Measurement of the Resistance of Electrolytes._--There are two effects
of the passage of an electric current which prevent the possibility of
measuring electrolytic resistance by the ordinary methods with the
direct currents which are used in the case of metals. The products of
the chemical decomposition of the electrolyte appear at the electrodes
and set up the opposing electromotive force of polarization, and unequal
dilution of the solution may occur in the neighbourhood of the two
electrodes. The chemical and electrolytic aspects of these phenomena are
treated in the article ELECTROLYSIS, but from our present point of view
also it is evident that they are again of fundamental importance. The
polarization at the surface of the electrodes will set up an opposing
electromotive force, and the unequal dilution of the solution will turn
the electrolyte into a concentration cell and produce a subsidiary
electromotive force either in the same direction as that applied or in
the reverse according as the anode or the cathode solution becomes the
more dilute. Both effects thus involve internal electromotive forces,
and prevent the application of Ohm's law to the electrolytic cell as a
whole. But the existence of a definite measurable resistance as a
characteristic property of the system depends on the conformity of the
system to Ohm's law, and it is therefore necessary to eliminate both
these effects before attempting to measure the resistance.

The usual and most satisfactory method of measuring the resistance of
electrolytes consists in eliminating the effects of polarization by the
use of alternating currents, that is, currents that are reversed in
direction many times a second.[11] The chemical action produced by the
first current is thus reversed by the second current in the opposite
direction, and the polarization caused by the first current on the
surface of the electrodes is destroyed before it rises to an appreciable
value. The polarization is also diminished in another way. The
electromotive force of polarization is due to the deposition of films of
the products of chemical decomposition on the surface of the electrodes,
and only reaches its full value when a continuous film is formed. If the
current be stopped before such a film is completed, the reverse
electromotive force is less than its full value. A given current flowing
for a given time deposits a definite amount of substance on the
electrodes, and therefore the amount per unit area is inversely
proportional to the area of the electrodes--to the area of contact, that
is, between the electrode and the liquid. Thus, by increasing the area
of the electrodes, the polarization due to a given current is decreased.
Now the area of free surface of a platinum plate can be increased
enormously by coating the plate with platinum black, which is metallic
platinum in a spongy state, and with such a plate as electrode the
effects of polarization are diminished to a very marked extent. The
coating is effected by passing an electric current first one way and
then the other between two platinum plates immersed in a 3% solution of
platinum chloride to which a trace of lead acetate is sometimes added.
The platinized plates thus obtained are quite satisfactory for the
investigation of strong solutions. They have the power, however, of
absorbing a certain amount of salt from the solutions and of giving it
up again when water or more dilute solution is placed in contact with
them. The measurement of very dilute solutions is thus made difficult,
but, if the plates be heated to redness after being platinized, a grey
surface is obtained which possesses sufficient area for use with dilute
solutions and yet does not absorb an appreciable quantity of salt.

Any convenient source of alternating current may be used. The currents
from the secondary circuit of a small induction coil are satisfactory,
or the currents of an alternating electric light supply may be
transformed down to an electromotive force of one or two volts. With
such currents it is necessary to consider the effects of self-induction
in the circuit and of electrostatic capacity. In balancing the
resistance of the electrolyte, resistance coils may be used in which
self-induction and the capacity are reduced to a minimum by winding the
wire of the coil backwards and forwards in alternate layers.

[Illustration: FIG. 1.]

With these arrangements the usual method of measuring resistance by
means of Wheatstone's bridge may be adapted to the case of electrolytes.
With alternating currents, however, it is impossible to use a
galvanometer in the usual way. The galvanometer was therefore replaced
by Kohlrausch by a telephone, which gives a sound when an alternating
current passes through it. The most common plan of the apparatus is
shown diagrammatically in fig. 1. The electrolytic cell and a resistance
box form two arms of the bridge, and the sliding contact is moved along
the metre wire which forms the other two arms till no sound is heard in
the telephone. The resistance of the electrolyte is to that of the box
as that of the right-hand end of the wire is to that of the left-hand
end. A more accurate method of using alternating currents, and one more
pleasant to use, gets rid of the telephone (_Phil. Trans._, 1900, 194,
p. 321). The current from one or two voltaic cells is led to an ebonite
drum turned by a motor or a hand-wheel and cord. On the drum are fixed
brass strips with wire brushes touching them in such a manner that the
current from the brushes is reversed several times in each revolution of
the drum. The wires from the brushes are connected with the Wheatstone's
bridge. A moving coil galvanometer is used as indicator, its connexions
being reversed in time with those of the battery by a slightly narrower
set of brass strips fixed on the other end of the ebonite commutator.
Thus any residual current through the galvanometer is direct and not
alternating. The high moment of inertia of the coil makes the period of
swing slow compared with the period of alternation of the current, and
the slight periodic disturbances are thus prevented from affecting the
galvanometer. When the measured resistance is not altered by increasing
the speed of the commutator or changing the ratio of the arms of the
bridge, the disturbing effects may be considered to be eliminated.

[Illustration: FIG. 2.]

[Illustration: FIG. 3.]

The form of vessel chosen to contain the electrolyte depends on the
order of resistance to be measured. For dilute solutions the shape of
cell shown in fig. 2 will be found convenient, while for more
concentrated solutions, that indicated in fig. 3 is suitable. The
absolute resistances of certain solutions have been determined by
Kohlrausch by comparison with mercury, and, by using one of these
solutions in any cell, the constant of that cell may be found once for
all. From the observed resistance of any given solution in the cell the
resistance of a centimetre cube--the so-called specific resistance--may
be calculated. The reciprocal of this, or the conductivity, is a more
generally useful constant; it is conveniently expressed in terms of a
unit equal to the reciprocal of an ohm. Thus Kohlrausch found that a
solution of potassium chloride, containing one-tenth of a gram
equivalent (7.46 grams) per litre, has at 18° C. a specific resistance
of 89.37 ohms per centimetre cube, or a conductivity of 1.119 × 10^-2
mhos or 1.119 × 10^-11 C.G.S. units. As the temperature variation of
conductivity is large, usually about 2% per degree, it is necessary to
place the resistance cell in a paraffin or water bath, and to observe
its temperature with some accuracy.

Another way of eliminating the effects of polarization and of dilution
has been used by W. Stroud and J. B. Henderson (_Phil. Mag._, 1897 [5],
43, p. 19). Two of the arms of a Wheatstone's bridge are composed of
narrow tubes filled with the solution, the tubes being of equal diameter
but of different length. The other two arms are made of coils of wire of
equal resistance, and metallic resistance is added to the shorter tube
till the bridge is balanced. Direct currents of somewhat high
electromotive force are used to work the bridge. Equal currents then
flow through the two tubes; the effects of polarization and dilution
must be the same in each, and the resistance added to the shorter tube
must be equal to the resistance of a column of liquid the length of
which is equal to the difference in length of the two tubes.

A somewhat different principle was adopted by E. Bouty in 1884. If a
current be passed through two resistances in series by means of an
applied electromotive force, the electric potential falls from one end
of the resistances to the other, and, if we apply Ohm's law to each
resistance in succession, we see that, since for each of them E = CR,
and C the current is the same through both, E the electromotive force or
fall of potential between the ends of each resistance must be
proportional to the resistance between them. Thus by measuring the
potential difference between the ends of the two resistances
successively, we may compare their resistances. If, on the other hand,
we can measure the potential difference in some known units, and
similarly measure the current flowing, we can determine the resistance
of a single electrolyte. The details of the apparatus may vary, but its
principle is illustrated in the following description. A narrow glass
tube is fixed horizontally into side openings in two glass vessels, and
an electric current passed through it by means of platinum electrodes
and a battery of considerable electromotive force. In this way a steady
fall of electric potential is set up along the length of the tube. To
measure the potential difference between the ends of the tube, tapping
electrodes are constructed, e.g. by placing zinc rods in vessels with
zinc sulphate solution and connecting these vessels (by means of thin
siphon tubes also filled with solution) with the vessels at the ends of
the long tube which contains the electrolyte to be examined. Whatever be
the contact potential difference between zinc and its solution, it is
the same at both ends, and thus the potential difference between the
zinc rods is equal to that between the liquid at the two ends of the
tube. This potential difference may be measured without passing any
appreciable current through the tapping electrodes, and thus the
resistance of the liquid deduced.

_Equivalent Conductivity of Solutions._--As is the case in the other
properties of solutions, the phenomena are much more simple when the
concentration is small than when it is great, and a study of dilute
solutions is therefore the best way of getting an insight into the
essential principles of the subject. The foundation of our knowledge was
laid by Kohlrausch when he had developed the method of measuring
electrolyte resistance described above. He expressed his results in
terms of "equivalent conductivity," that is, the conductivity (k) of the
solution divided by the number (m) of gram-equivalents of electrolyte
per litre. He finds that, as the concentration diminishes, the value of
k/m approaches a limit, and eventually becomes constant, that is to say,
at great dilution the conductivity is proportional to the concentration.
Kohlrausch first prepared very pure water by repeated distillation and
found that its resistance continually increased as the process of
purification proceeded. The conductivity of the water, and of the slight
impurities which must always remain, was subtracted from that of the
solution made with it, and the result, divided by m, gave the equivalent
conductivity of the substance dissolved. This procedure appears
justifiable, for as long as conductivity is proportional to
concentration it is evident that each part of the dissolved matter
produces its own independent effect, so that the total conductivity is
the sum of the conductivities of the parts; when this ceases to hold,
the concentration of the solution has in general become so great that
the conductivity of the solvent may be neglected. The general result of
these experiments can be represented graphically by plotting k/m as
ordinates and [root 3]m as abscissae, [root 3]m being a number
proportional to the reciprocal of the average distance between the
molecules, to which it seems likely that the molecular conductivity may
be related. The general types of curve for a simple neutral salt like
potassium or sodium chloride and for a caustic alkali or acid are shown
in fig. 4. The curve for the neutral salt comes to a limiting value;
that for the acid attains a maximum at a certain very small
concentration, and falls again when the dilution is carried farther. It
has usually been considered that this destruction of conductivity is due
to chemical action between the acid and the residual impurities in the
water. At such great dilution these impurities are present in quantities
comparable with the amount of acid which they convert into a less highly
conducting neutral salt. In the case of acids, then, the maximum must be
taken as the limiting value. The decrease in equivalent conductivity at
great dilution is, however, so constant that this explanation seems
insufficient. The true cause of the phenomenon may perhaps be connected
with the fact that the bodies in which it occurs, acids and alkalis,
contain the ions, hydrogen in the one case, hydroxyl in the other, which
are present in the solvent, water, and have, perhaps because of this
relation, velocities higher than those of any other ions. The values of
the molecular conductivities of all neutral salts are, at great
dilution, of the same order of magnitude, while those of acids at their
maxima are about three times as large. The influence of increasing
concentration is greater in the case of salts containing divalent ions,
and greatest of all in such cases as solutions of ammonia and acetic
acid, which are substances of very low conductivity.

[Illustration: FIG. 4.]

_Theory of Moving Ions._--Kohlrausch found that, when the polarization
at the electrodes was eliminated, the resistance of a solution was
constant however determined, and thus established Ohm's Law for
electrolytes. The law was confirmed in the case of strong currents by G.
F. Fitzgerald and F. T. Trouton (_B.A. Report_, 1886, p. 312). Now,
Ohm's Law implies that no work is done by the current in overcoming
reversible electromotive forces such as those of polarization. Thus the
molecular interchange of ions, which must occur in order that the
products may be able to work their way through the liquid and appear at
the electrodes, continues throughout the solution whether a current is
flowing or not. The influence of the current on the ions is merely
directive, and, when it flows, streams of electrified ions travel in
opposite directions, and, if the applied electromotive force is enough
to overcome the local polarization, give up their charges to the
electrodes. We may therefore represent the facts by considering the
process of electrolysis to be a kind of convection. Faraday's classical
experiments proved that when a current flows through an electrolyte the
quantity of substance liberated at each electrode is proportional to its
chemical equivalent weight, and to the total amount of electricity
passed. Accurate determinations have since shown that the mass of an ion
deposited by one electromagnetic unit of electricity, i.e. its
electro-chemical equivalent, is 1.036 × 10^-4 × its chemical equivalent
weight. Thus the amount of electricity associated with one
gram-equivalent of any ion is 10^4/1.036 = 9653 units. Each monovalent
ion must therefore be associated with a certain definite charge, which
we may take to be a natural unit of electricity; a divalent ion carries
two such units, and so on. A cation, i.e. an ion giving up its charge at
the cathode, as the electrode at which the current leaves the solution
is called, carries a positive charge of electricity; an anion,
travelling in the opposite direction, carries a negative charge. It will
now be seen that the quantity of electricity flowing per second, i.e.
the current through the solution, depends on (1) the number of the ions
concerned, (2) the charge on each ion, and (3) the velocity with which
the ions travel past each other. Now, the number of ions is given by the
concentration of the solution, for even if all the ions are not actively
engaged in carrying the current at the same instant, they must, on any
dynamical idea of chemical equilibrium, be all active in turn. The
charge on each, as we have seen, can be expressed in absolute units, and
therefore the velocity with which they move past each other can be
calculated. This was first done by Kohlrausch (_Göttingen Nachrichten_,
1876, p. 213, and _Das Leitvermögen der Elektrolyte_, Leipzig, 1898)
about 1879.

  In order to develop Kohlrausch's theory, let us take, as an example,
  the case of an aqueous solution of potassium chloride, of
  concentration n gram-equivalents per cubic centimetre. There will then
  be n gram-equivalents of potassium ions and the same number of
  chlorine ions in this volume. Let us suppose that on each
  gram-equivalent of potassium there reside +e units of electricity, and
  on each gram-equivalent of chlorine ions -e units. If u denotes the
  average velocity of the potassium ion, the positive charge carried per
  second across unit area normal to the flow is n e u. Similarly, if v
  be the average velocity of the chlorine ions, the negative charge
  carried in the opposite direction is n e v. But positive electricity
  moving in one direction is equivalent to negative electricity moving
  in the other, so that, before changes in concentration sensibly
  supervene, the total current, C, is ne(u + v). Now let us consider the
  amounts of potassium and chlorine liberated at the electrodes by this
  current. At the cathode, if the chlorine ions were at rest, the excess
  of potassium ions would be simply those arriving in one second,
  namely, nu. But since the chlorine ions move also, a further
  separation occurs, and nv potassium ions are left without partners.
  The total number of gram-equivalents liberated is therefore n(u + v).
  By Faraday's law, the number of grams liberated is equal to the
  product of the current and the electro-chemical equivalent of the ion;
  the number of gram-equivalents therefore must be equal to [eta]C,
  where [eta] denotes the electro-chemical equivalent of hydrogen in
  C.G.S. units. Thus we get

    n(u + v) = [eta]C = [eta]ne(u + v),

  and it follows that the charge, e, on 1 gram-equivalent of each kind
  of ion is equal to 1/[eta]. We know that Ohm's Law holds good for
  electrolytes, so that the current C is also given by k·dP/dx, where k
  denotes the conductivity of the solution, and dP/dx the potential
  gradient, i.e. the change in potential per unit length along the lines
  of current flow. Thus

      n
    -----(u + v) = kdP/dx;
    [eta]

  therefore

                  k dP
    u + v = [eta] - --.
                  n dx

  Now [eta] is 1.036 × 10^-4, and the concentration of a solution is
  usually expressed in terms of the number, m, of gram-equivalents per
  litre instead of per cubic centimetre. Therefore

                            k  dP
    u + v = 1.036 × 10^-1 - -- --.
                            m  dx

  When the potential gradient is one volt (10^8 C.G.S. units) per
  centimetre this becomes

    u + v = 1.036 × 10^-7 × k/m.

  Thus by measuring the value of k/m, which is known as the equivalent
  conductivity of the solution, we can find u + v, the velocity of the
  ions relative to each other. For instance, the equivalent conductivity
  of a solution of potassium chloride containing one-tenth of a
  gram-equivalent per litre is 1119 × 10^-13 C.G.S. units at 18° C.
  Therefore

      u + v = 1.036 × 10^7 × 1119 × 10^-13
    = 1.159 × 10^-3 = 0.001159 cm. per sec.

  In order to obtain the absolute velocities u and v, we must find some
  other relation between them. Let us resolve u into ½(u + v) in one
  direction, say to the right, and ½(u - v) to the left. Similarly v can
  be resolved into ½(v+u) to the left and ½(v-u) to the right. On
  pairing these velocities we have a combined movement of the ions to
  the right, with a speed of ½(u - v) and a drift right and left, past
  each other, each ion travelling with a speed of ½(u + v), constituting
  the electrolytic separation. If u is greater than v, the combined
  movement involves a concentration of salt at the cathode, and a
  corresponding dilution at the anode, and _vice versa_. The rate at
  which salt is electrolysed, and thus removed from the solution at each
  electrode, is ½(u + v). Thus the total loss of salt at the cathode is
  ½(u + v) - ½(u - v) or v, and at the anode, ½(v + u) - ½(v - u), or u.
  Therefore, as is explained in the article ELECTROLYSIS, by measuring
  the dilution of the liquid round the electrodes when a current passed,
  W. Hittorf (_Pogg. Ann._, 1853-1859, 89, p. 177; 98, p. 1; 103, p. 1;
  106, pp. 337 and 513) was able to deduce the ratio of the two
  velocities, for simple salts when no complex ions are present, and
  many further experiments have been made on the subject (see _Das
  Leitvermögen der Elektrolyte_).

  By combining the results thus obtained with the sum of the velocities,
  as determined from the conductivities, Kohlrausch calculated the
  absolute velocities of different ions under stated conditions. Thus,
  in the case of the solution of potassium chloride considered above,
  Hittorf's experiments show us that the ratio of the velocity of the
  anion to that of the cation in this solution is .51 : .49. The
  absolute velocity of the potassium ion under unit potential gradient
  is therefore 0.000567 cm. per sec., and that of the chlorine ion
  0.000592 cm. per sec. Similar calculations can be made for solutions
  of other concentrations, and of different substances.

Table IX. shows Kohlrausch's values for the ionic velocities of three
chlorides of alkali metals at 18° C, calculated for a potential gradient
of 1 volt per cm.; the numbers are in terms of a unit equal to 10^-6 cm.
per sec.:--

  TABLE IX.

  +--------------------------------------------+----------------+
  |         |       KCl       |      NaCl      |      LiCl      |
  +---------+-----------------+----------------+----------------+
  |    m    | u + v  u    v   | u + v   u   v  |u + v   u    v  |
  +---------+-----------------+----------------+----------------+
  |  0      | 1350  660  690  | 1140  450  690 | 1050  360  690 |
  |  0.0001 | 1335  654  681  | 1129  448  681 | 1037  356  681 |
  |   .001  | 1313  643  670  | 1110  440  670 | 1013  343  670 |
  |   .01   | 1263  619  644  | 1059  415  644 |  962  318  644 |
  |   .03   | 1218  597  621  | 1013  390  623 |  917  298  619 |
  |   .1    | 1153  564  589  |  952  360  592 |  853  259  594 |
  |   .3    | 1088  531  557  |  876  324  552 |  774  217  557 |
  |  1.0    | 1011  491  520  |  765  278  487 |  651  169  482 |
  |  3.0    |  911  442  469  |  582  206  376 |  463  115  348 |
  |  5.0    |                 |  438  153  285 |  334   80  254 |
  | 10.0    |                 |                |  117   25   92 |
  +---------+-----------------+----------------+----------------+

These numbers show clearly that there is an increase in ionic velocity
as the dilution proceeds. Moreover, if we compare the values for the
chlorine ion obtained from observations on these three different salts,
we see that as the concentrations diminish the velocity of the chlorine
ion becomes the same in all of them. A similar relation appears in other
cases, and, in general, we may say that at great dilution the velocity
of an ion is independent of the nature of the other ion present. This
introduces the conception of specific ionic velocities, for which some
values at 18° C. are given by Kohlrausch in Table X.:--

  Table X.

  +------------------------------------+
  | K         66 × 10^-5 cms. per sec. |
  | Na        45          "      "     |
  | Li        36          "      "     |
  | NH4       66          "      "     |
  | H        320          "      "     |
  | Ag        57          "      "     |
  | Cl        69          "      "     |
  | I         69          "      "     |
  | NO3       64          "      "     |
  | OH       162          "      "     |
  | C2H3O2    36          "      "     |
  | C3H5O2    33          "      "     |
  +------------------------------------+

Having obtained these numbers we can deduce the conductivity of the
dilute solution of any salt, and the comparison of the calculated with
the observed values furnished the first confirmation of Kohlrausch's
theory. Some exceptions, however, are known. Thus acetic acid and
ammonia give solutions of much lower conductivity than is indicated by
the sum of the specific ionic velocities of their ions as determined
from other compounds. An attempt to find in Kohlrausch's theory some
explanation of this discrepancy shows that it could be due to one of two
causes. Either the velocities of the ions must be much less in these
solutions than in others, or else only a fractional part of the number
of molecules present can be actively concerned in conveying the current.
We shall return to this point later.

  _Friction on the Ions._--It is interesting to calculate the magnitude
  of the forces required to drive the ions with a certain velocity. If
  we have a potential gradient of 1 volt per centimetre the electric
  force is 10^8 in C.G.S. units. The charge of electricity on 1
  gram-equivalent of any ion is 1/.0001036 = 9653 units, hence the
  mechanical force acting on this mass is 9653 × 10^8 dynes. This, let
  us say, produces a velocity u; then the force required to produce unit
  velocity is PA = 9.653 × 10^11/u dynes = 9.84 × 10^5/u kilograms-weight.
  If the ion have an equivalent weight A, the force producing unit
  velocity when acting on 1 gram is P1 = 9.84 × 10^5/Au kilograms-weight.
  Thus the aggregate force required to drive 1 gram of potassium ions
  with a velocity of 1 centimetre per second through a very dilute
  solution must be equal to the weight of 38 million kilograms.

    Table XI.

    +-------------------------------+
    |       Kilograms-weight.       |
    +-------------------------------+
    |            PA          P1     |
    |K        15 × 10^8   38 × 10^6 |
    |Na       22    "      95   "   |
    |Li       27    "     390   "   |
    |NH4      15    "      83   "   |
    |H         3.1  "     310   "   |
    |Ag       17    "      16   "   |
    |Cl       14    "      40   "   |
    |I        14    "      11   "   |
    |NO3      15    "      25   "   |
    |OH        5.4  "      32   "   |
    |C2H8O2   27    "      46   "   |
    |C3H5O2   30    "      41   "   |
    +------------- -----------------+

  Since the ions move with uniform velocity, the frictional resistances
  brought into play must be equal and opposite to the driving forces,
  and therefore these numbers also represent the ionic friction
  coefficients in very dilute solutions at 18° C.

_Direct Measurement of Ionic Velocities._--Sir Oliver Lodge was the
first to directly measure the velocity of an ion (_B.A. Report_, 1886,
p. 389). In a horizontal glass tube connecting two vessels filled with
dilute sulphuric acid he placed a solution of sodium chloride in solid
agar-agar jelly. This solid solution was made alkaline with a trace of
caustic soda in order to bring out the red colour of a little
phenol-phthalein added as indicator. An electric current was then passed
from one vessel to the other. The hydrogen ions from the anode vessel of
acid were thus carried along the tube, and, as they travelled,
decolourized the phenol-phthalein. By this method the velocity of the
hydrogen ion through a jelly solution under a known potential gradient
was observed to about 0.0026 cm. per sec, a number of the same order as
that required by Kohlrausch's theory. Direct determinations of the
velocities of a few other ions have been made by W. C. D. Whetham
(_Phil. Trans._ vol. 184, A, p. 337; vol. 186, A, p. 507; _Phil. Mag._,
October 1894). Two solutions having one ion in common, of equivalent
concentrations, different densities, different colours, and nearly equal
specific resistances, were placed one over the other in a vertical glass
tube. In one case, for example, decinormal solutions of potassium
carbonate and potassium bichromate were used. The colour of the latter
is due to the presence of the bichromate group, Cr2O7. When a current
was passed across the junction, the anions CO3 and Cr2O7 travelled in
the direction opposite to that of the current, and their velocity could
be determined by measuring the rate at which the colour boundary moved.
Similar experiments were made with alcoholic solutions of cobalt salts,
in which the velocities of the ions were found to be much less than in
water. The behaviour of agar jelly was then investigated, and the
velocity of an ion through a solid jelly was shown to be very little
less than in an ordinary liquid solution. The velocities could therefore
be measured by tracing the change in colour of an indicator or the
formation of a precipitate. Thus decinormal jelly solutions of barium
chloride and sodium chloride, the latter containing a trace of sodium
sulphate, were placed in contact. Under the influence of an
electromotive force the barium ions moved up the tube, disclosing their
presence by the trace of insoluble barium sulphate formed. Again, a
measurement of the velocity of the hydrogen ion, when travelling through
the solution of an acetate, showed that its velocity was then only about
the one-fortieth part of that found during its passage through
chlorides. From this, as from the measurements on alcohol solutions, it
is clear that where the equivalent conductivities are very low the
effective velocities of the ions are reduced in the same proportion.

Another series of direct measurements has been made by Orme Masson
(_Phil. Trans._ vol. 192, A, p. 331). He placed the gelatine solution of
a salt, potassium chloride, for example, in a horizontal glass tube, and
found the rate of migration of the potassium and chlorine ions by
observing the speed at which they were replaced when a coloured anion,
say, the Cr2O7 from a solution of potassium bichromate, entered the tube
at one end, and a coloured cation, say, the Cu from copper sulphate, at
the other. The coloured ions are specifically slower than the colourless
ions which they follow, and in this case it follows that the coloured
solution has a higher resistance than the colourless. For the same
current, therefore, the potential gradient is higher in the coloured
solution and lower in the colourless one. Thus a coloured ion which gets
in front of the advancing boundary finds itself acted on by a smaller
force and falls back into line, while a straggling colourless ion is
pushed forward again. Hence a sharp boundary is preserved. B. D. Steele
has shown that with these sharp boundaries the use of coloured ions is
unnecessary, the junction line being visible owing to the difference in
the optical refractive indices of two colourless solutions. Once the
boundary is formed, too, no gelatine is necessary, and the motion can be
watched through liquid aqueous solutions (see R. B. Denison and B. D.
Steele, _Phil. Trans._, 1906).

All the direct measurements which have been made on simple binary
electrolytes agree with Kohlrausch's results within the limits of
experimental error. His theory, therefore, probably holds good in such
cases, whatever be the solvent, if the proper values are given to the
ionic velocities, i.e. the values expressing the velocities with which
the ions actually move in the solution of the strength taken, and under
the conditions of the experiment. If we know the specific velocity of
any one ion, we can deduce, from the conductivity of very dilute
solutions, the velocity of any other ion with which it may be
associated, a proceeding which does not involve the difficult task of
determining the migration constant of the compound. Thus, taking the
specific ionic velocity of hydrogen as 0.00032 cm. per second, we can
find, by determining the conductivity of dilute solutions of any acid,
the specific velocity of the acid radicle involved. Or again, since we
know the specific velocity of silver, we can find the velocities of a
series of acid radicles at great dilution by measuring the conductivity
of their silver salts.

  By such methods W. Ostwald, G. Bredig and other observers have found
  the specific velocities of many ions both of inorganic and organic
  compounds, and examined the relation between constitution and ionic
  velocity. The velocity of elementary ions is found to be a periodic
  function of the atomic weight, similar elements lying on corresponding
  portions of a curve drawn to express the relation between these two
  properties. Such a curve much resembles that giving the relation
  between atomic weight and viscosity in solution. For complex ions the
  velocity is largely an additive property; to a continuous additive
  change in the composition of the ion corresponds a continuous but
  decreasing change in the velocity. The following table gives Ostwald's
  results for the formic acid  series:--

    Table XII.

    +----------------------+----------+---------------------+
    |                      | Velocity.| Difference for CH2. |
    +----------------------+----------+---------------------+
    | Formic acid  HCO2    |   51.2   |           ..        |
    | Acetic    "  H3C2O2  |   38.3   |        -12.9        |
    | Propionic "  H5C3O2  |   34.3   |        - 4.0        |
    | Butyric   "  H7C4O2  |   30.8   |        - 3.5        |
    | Valeric   "  H9C5O2  |   28.8   |        - 2.0        |
    | Caprionic "  H11C6O2 |   27.4   |        - 1.4        |
    +----------------------+----------+---------------------+

_Nature of Electrolytes._--We have as yet said nothing about the
fundamental cause of electrolytic activity, nor considered why, for
example, a solution of potassium chloride is a good conductor, while a
solution of sugar allows practically no current to pass.

All the preceding account of the subject is, then, independent of any
view we may take of the nature of electrolytes, and stands on the basis
of direct experiment. Nevertheless, the facts considered point to a very
definite conclusion. The specific velocity of an ion is independent of
the nature of the opposite ion present, and this suggests that the ions
themselves, while travelling through the liquid, are dissociated from
each other. Further evidence, pointing in the same direction, is
furnished by the fact that since the conductivity is proportional to the
concentration at great dilution, the equivalent-conductivity, and
therefore the ionic velocity, is independent of it. The importance of
this relation will be seen by considering the alternative to the
dissociation hypothesis. If the ions are not permanently free from each
other their mobility as parts of the dissolved molecules must be secured
by continual interchanges. The velocity with which they work their way
through the liquid must then increase as such molecular rearrangements
become more frequent, and will therefore depend on the number of solute
molecules, i.e. on the concentration. On this supposition the observed
constancy of velocity would be impossible. We shall therefore adopt as a
wording hypothesis the theory, confirmed by other phenomena (see
ELECTROLYSIS), that an electrolyte consists of dissociated ions.

It will be noticed that neither the evidence in favour of the
dissociation theory which is here considered, nor that described in the
article ELECTROLYSIS, requires more than the effective dissociation of
the ions from each other. They may well be connected in some way with
solvent molecules, and there are several indications that an ion
consists of an electrified part of the molecule of the dissolved salt
with an attendant atmosphere of solvent round it. The conductivity of a
salt solution depends on two factors--(1) the fraction of the salt
ionized; (2) the velocity with which the ions, when free from each
other, move under the electric forces.[12] When a solution is heated,
both these factors may change. The coefficient of ionization usually,
though not always, decreases; the specific ionic velocities increase.
Now the rate of increase with temperature of these ionic velocities is
very nearly identical with the rate of decrease of the viscosity of the
liquid. If the curves obtained by observations at ordinary temperatures
be carried on they indicate a zero of fluidity and a zero of ionic
velocity about the same point, 38.5° C. below the freezing point of
water (Kohlrausch, _Sitz. preuss. Akad. Wiss._, 1901, 42, p. 1026). Such
relations suggest that the frictional resistance to the motion of an ion
is due to the ordinary viscosity of the liquid, and that the ion is
analogous to a body of some size urged through a viscous medium rather
than to a particle of molecular dimensions finding its way through a
crowd of molecules of similar magnitude. From this point of view W. K.
Bousfield has calculated the sizes of ions on the assumption that
Stokes's theory of the motion of a small sphere through a viscous medium
might be applied (_Zeits. phys. Chem._, 1905, 53, p. 257; _Phil. Trans._
A, 1906, 206, p. 101). The radius of the potassium or chlorine ion with
its envelope of water appears to be about 1.2 × 10^-8 centimetres.

  For the bibliography of electrolytic conduction see ELECTROLYSIS. The
  books which deal more especially with the particular subject of the
  present article are _Das Leitvermögen der Elektrolyte_, by F.
  Kohlrausch and L. Holborn (Leipzig, 1898), and _The Theory of Solution
  and Electrolysis_, by W. C. D. Whetham (Cambridge, 1902).
        (W. C. D. W.)


III. ELECTRIC CONDUCTION THROUGH GASES

A gas such as air when it is under normal conditions conducts
electricity to a small but only to a very small extent, however small
the electric force acting on the gas may be. The electrical conductivity
of gases not exposed to special conditions is so small that it was only
definitely established in the early years of the 20th century, although
it had engaged the attention of physicists for more than a hundred
years. It had been known for a long time that a body charged with
electricity slowly lost its charge even when insulated with the greatest
care, and though long ago some physicists believed that part of the leak
of electricity took place through the air, the general view seems to
have been that it was due to almost unavoidable defects in the
insulation or to dust in the air, which after striking the charged body
was repelled from it and went off with some of the charge. C. A.
Coulomb, who made some very careful experiments which were published in
1785 (_Mém. de l'Acad. des Sciences_, 1785, p. 612), came to the
conclusion that after allowing for the leakage along the threads which
supported the charged body there was a balance over, which he attributed
to leakage through the air. His view was that when the molecules of air
come into contact with a charged body some of the electricity goes on to
the molecules, which are then repelled from the body carrying their
charge with them. We shall see later that this explanation is not
tenable. C. Matteucci (_Ann. chim. phys._, 1850, 28, p. 390) in 1850
also came to the conclusion that the electricity from a charged body
passes through the air; he was the first to prove that the rate at
which electricity escapes is less when the pressure of the gas is low
than when it is high. He found that the rate was the same whether the
charged body was surrounded by air, carbonic acid or hydrogen.
Subsequent investigations have shown that the rate in hydrogen is in
general much less than in air. Thus in 1872 E. G. Warburg (_Pogg. Ann._,
1872, 145, p. 578) found that the leak through hydrogen was only about
one-half of that through air: he confirmed Matteucci's observations on
the effect of pressure on the rate of leak, and also found that it was
the same whether the gas was dry or damp. He was inclined to attribute
the leak to dust in the air, a view which was strengthened by an
experiment of J. W. Hittorf's (_Wied. Ann._, 1879, 7, p. 595), in which
a small carefully insulated electroscope, placed in a small vessel
filled with carefully filtered gas, retained its charge for several
days; we know now that this was due to the smallness of the vessel and
not to the absence of dust, as it has been proved that the rate of leak
in small vessels is less than in large ones.

Great light was thrown on this subject by some experiments on the rates
of leak from charged bodies in closed vessels made almost simultaneously
by H. Geitel (_Phys. Zeit._, 1900, 2, p. 116) and C. T. R. Wilson
(_Proc. Camb. Phil._ Soc., 1900, 11, p. 32). These observers established
that (1) the rate of escape of electricity in a closed vessel is much
smaller than in the open, and the larger the vessel the greater is the
rate of leak; and (2) the rate of leak does not increase in proportion
to the differences of potential between the charged body and the walls
of the vessel: the rate soon reaches a limit beyond which it does not
increase, however much the potential difference may be increased,
provided, of course, that this is not great enough to cause sparks to
pass from the charged body. On the assumption that the maximum leak is
proportional to the volume, Wilson's experiments, which were made in
vessels less than 1 litre in volume, showed that in dust-free air at
atmospheric pressure the maximum quantity of electricity which can
escape in one second from a charged body in a closed volume of V cubic
centimetres is about 10^-8V electrostatic units. E. Rutherford and S. T.
Allan (_Phys. Zeit._, 1902, 3, p. 225), working in Montreal, obtained
results in close agreement with this. Working between pressures of from
43 to 743 millimetres of mercury, Wilson showed that the maximum rate of
leak is very approximately proportional to the pressure; it is thus
exceedingly small when the pressure is low--a result illustrated in a
striking way by an experiment of Sir W. Crookes (_Proc. Roy. Soc._,
1879, 28, p. 347) in which a pair of gold leaves retained an electric
charge for several months in a very high vacuum. Subsequent experiments
have shown that it is only in very small vessels that the rate of leak
is proportional to the volume and to the pressure; in large vessels the
rate of leak per unit volume is considerably smaller than in small ones.
In small vessels the maximum rate of leak in different gases, is, with
the exception of hydrogen, approximately proportional to the density of
the gas. Wilson's results on this point are shown in the following table
(Proc. Roy. Soc., 1901, 60, p. 277):--

  +---------+------------------------+-----------------+
  |   Gas.  | Relative Rate of Leak. | _Rate of Leak._ |
  |         |                        |       Sp. Gr.   |
  +---------+------------------------+-----------------+
  | Air     |          1.00          |       1         |
  | H2      |           .184         |       2.7       |
  | CO2     |          1.69          |       1.10      |
  | SO2     |          2.64          |       1.21      |
  | CH3Cl   |          4.7           |       1.09      |
  | Ni(CO)4 |          5.1           |        .867     |
  +---------+------------------------+-----------------+

The rate of leak of electricity through gas contained in a closed vessel
depends to some extent on the material of which the walls of the vessel
are made; thus it is greater, other circumstances being the same, when
the vessel is made of lead than when it is made of aluminium. It also
varies, as Campbell and Wood (_Phil. Mag._ [6], 13, p. 265) have shown,
with the time of the day, having a well-marked minimum at about 3
o'clock in the morning: it also varies from month to month. Rutherford
(_Phys. Rev._, 1903, 16, p. 183), Cooke (_Phil. Mag._, 1903 [6], 6, p.
403) and M'Clennan and Burton (_Phys. Rev._, 1903, 16, p. 184) have
shown that the leak in a closed vessel can be reduced by about 30% by
surrounding the vessel with sheets of thick lead, but that the reduction
is not increased beyond this amount, however thick the lead sheets may
be. This result indicates that part of the leak is due to a very
penetrating kind of radiation, which can get through the thin walls of
the vessel but is stopped by the thick lead. A large part of the leak we
are describing is due to the presence of radioactive substances such as
radium and thorium in the earth's crust and in the walls of the vessel,
and to the gaseous radioactive emanations which diffuse from them into
the atmosphere. This explains the very interesting effect discovered by
J. Elster and H. Geitel (_Phys. Zeit._, 1901, 2, p. 560), that the rate
of leak in caves and cellars when the air is stagnant and only renewed
slowly is much greater than in the open air. In some cases the
difference is very marked; thus they found that in the cave called the
Baumannshöhle in the Harz mountains the electricity escaped at seven
times the rate it did in the air outside. In caves and cellars the
radioactive emanations from the walls can accumulate and are not blown
away as in the open air.

The electrical conductivity of gases in the normal state is, as we have
seen, exceedingly small, so small that the investigation of its
properties is a matter of considerable difficulty; there are, however,
many ways by which the electrical conductivity of a gas can be increased
so greatly that the investigation becomes comparatively easy. Among such
methods are raising the temperature of the gas above a certain point.
Gases drawn from the neighbourhood of flames, electric arcs and sparks,
or glowing pieces of metal or carbon are conductors, as are also gases
through which Röntgen or cathode rays or rays of positive electricity
are passing; the rays from the radioactive metals, radium, thorium,
polonium and actinium, produce the same effect, as does also
ultra-violet light of exceedingly short wave-length. The gas, after
being made a conductor of electricity by any of these means, is found to
possess certain properties; thus it retains its conductivity for some
little time after the agent which made it a conductor has ceased to act,
though the conductivity diminishes very rapidly and finally gets too
small to be appreciable.

[Illustration: FIG. 5.]

This and several other properties of conducting gas may readily be
proved by the aid of the apparatus represented in fig. 5. V is a testing
vessel in which an electroscope is placed. Two tubes A and C are fitted
into the vessel, A being connected with a water pump, while the far end
of C is in the region where the gas is exposed to the agent which makes
it a conductor of electricity. Let us suppose that the gas is made
conducting by Röntgen rays produced by a vacuum tube which is placed in
a box, covered except for a window at B with lead so as to protect the
electroscope from the direct action of the rays. If a slow current of
air is drawn by the water pump through the testing vessel, the charge on
the electroscope will gradually leak away. The leak, however, ceases
when the current of air is stopped. This result shows that the gas
retains its conductivity during the time taken by it to pass from one
end to the other of the tube C.

The gas loses its conductivity when filtered through a plug of
glass-wool, or when it is made to bubble through water. This can readily
be proved by inserting in the tube C a plug of glass-wool or a water
trap; then if by working the pump a little harder the same current of
air is produced as before, it will be found that the electroscope will
now retain its charge, showing that the conductivity can, as it were, be
filtered out of the gas. The conductivity can also be removed from the
gas by making the gas traverse a strong electric field. We can show this
by replacing the tube C by a metal tube with an insulated wire passing
down the axis of the tube. If there is no potential difference between
the wire and the tube then the electroscope will leak when a current of
air is drawn through the vessel, but the leak will stop if a
considerable difference of potential is maintained between the wire and
the tube: this shows that a strong electric field removes the
conductivity from the gas.

The fact that the conductivity of the gas is removed by filtering shows
that it is due to something mixed with the gas which is removed from it
by filtration, and since the conductivity is also removed by an electric
field, the cause of the conductivity must be charged with electricity so
as to be driven to the sides of the tube by the electric force. Since
the gas as a whole is not electrified either positively or negatively,
there must be both negative and positive charges in the gas, the amount
of electricity of one sign being equal to that of the other. We are thus
led to the conclusion that the conductivity of the gas is due to
electrified particles being mixed up with the gas, some of these
particles having charges of positive electricity, others of negative.
These electrified particles are called _ions_, and the process by which
the gas is made a conductor is called the ionization of the gas. We
shall show later that the charges and masses of the ions can be
determined, and that the gaseous ions are not identical with those met
with in the electrolysis of solutions.

[Illustration: FIG. 6.]

One very characteristic property of conduction of electricity through a
gas is the relation between the current through the gas and the electric
force which gave rise to it. This relation is not in general that
expressed by Ohm's law, which always, as far as our present knowledge
extends, expresses the relation for conduction through metals and
electrolytes. With gases, on the other hand, it is only when the current
is very small that Ohm's law is true. If we represent graphically by
means of a curve the relation between the current passing between two
parallel metal plates separated by ionized gas and the difference of
potential between the plates, the curve is of the character shown in
fig. 6 when the ordinates represent the current and the abscissae the
difference of potential between the plates. We see that when the
potential difference is very small, i.e. close to the origin, the curve
is approximately straight, but that soon the current increases much less
rapidly than the potential difference, and that a stage is reached when
no appreciable increase of current is produced when the potential
difference is increased; when this stage is reached the current is
constant, and this value of the current is called the "saturation"
value. When the potential difference approaches the value at which
sparks would pass through the gas, the current again increases with the
potential difference; thus the curve representing the relation between
the current and potential difference over very wide ranges of potential
difference has the shape shown in fig. 7; curves of this kind have been
obtained by von Schweidler (_Wien. Ber._, 1899, 108, p. 273), and J. E.
S. Townsend (_Phil. Mag._, 1901 [6], 1, p. 198). We shall discuss later
the causes of the rise in the current with large potential differences,
when we consider ionization by collision.

  The general features of the earlier part of the curve are readily
  explained on the ionization hypothesis. On this view the Röntgen rays
  or other ionizing agent acting on the gas between the plates, produces
  positive and negative ions at a definite rate. Let us suppose that q
  positive and q negative ions are by this means produced per second
  between the plates; these under the electric force will tend to move,
  the positive ones to the negative plate, the negative ones to the
  positive. Some of these ions will reach the plate, others before
  reaching the plate will get so near one of the opposite sign that the
  attraction between them will cause them to unite and form an
  electrically neutral system; when they do this they end their
  existence as ions. The current between the plates is proportional to
  the number of ions which reach the plates per second. Now it is
  evident that we cannot go on taking more ions out of the gas than are
  produced; thus we cannot, when the current is steady, have more than q
  positive ions driven to the negative plate per second, and the same
  number of negative ions to the positive. If each of the positive ions
  carries a charge of e units of positive electricity, and if there is
  an equal and opposite charge on each negative ion, then the maximum
  amount of electricity which can be given to the plates per second is
  qe, and this is equal to the saturation current. Thus if we measure
  the saturation current, we get a direct measure of the ionization, and
  this does not require us to know the value of any quantity except the
  constant charge on the ion. If we attempted to deduce the amount of
  ionization by measurements of the current before it was saturated, we
  should require to know in addition the velocity with which the ions
  move under a given electric force, the time that elapses between the
  liberation of an ion and its combination with one of the opposite
  sign, and the potential difference between the plates. Thus if we wish
  to measure the amount of ionization in a gas we should be careful to
  see that the current is saturated.

  [Illustration: FIG. 7.]

The difference between conduction through gases and through metals is
shown in a striking way when we use potential differences large enough
to produce the saturation current. Suppose we have got a potential
difference between the plates more than sufficient to produce the
saturation current, and let us increase the distance between the plates.
If the gas were to act like a metallic conductor this would diminish the
current, because the greater length would involve a greater resistance
in the circuit. In the case we are considering the separation of the
plates will _increase_ the current, because now there is a larger volume
of gas exposed to the rays; there are therefore more ions produced, and
as the saturation current is proportional to the number of ions the
saturation current is increased. If the potential difference between the
plates were much less than that required to saturate the current, then
increasing the distance would diminish the current; the gas for such
potential differences obeys Ohm's law and the behaviour of the gaseous
resistance is therefore similar to that of a metallic one.

In order to produce the saturation current the electric field must be
strong enough to drive each ion to the electrode before it has time to
enter into combination with one of the opposite sign. Thus when the
plates in the preceding example are far apart, it will take a larger
potential difference to produce this current than when the plates are
close together. The potential difference required to saturate the
current will increase as the square of the distance between the plates,
for if the ions are to be delivered in a given time to the plates their
speed must be proportional to the distance between the plates. But the
speed is proportional to the electric force acting on the ion; hence the
electric force must be proportional to the distance between the plates,
and as in a uniform field the potential difference is equal to the
electric force multiplied by the distance between the plates, the
potential difference will vary as the square of this distance.

The potential difference required to produce saturation will, other
circumstances being the same, increase with the amount of ionization,
for when the number of ions is large and they are crowded together, the
time which will elapse before a positive one combines with a negative
will be smaller than when the number of ions is small. The ions have
therefore to be removed more quickly from the gas when the ionization is
great than when it is small; thus they must move at a higher speed and
must therefore be acted upon by a larger force.


When the ions are not removed from the gas, they will increase until the
number of ions of one sign which combine with ions of the opposite sign
in any time is equal to the number produced by the ionizing agent in
that time. We can easily calculate the number of free ions at any time
after the ionizing agent has commenced to act.

  Let q be the number of ions (positive or negative) produced in one
  cubic centimetre of the gas per second by the ionizing agent, n1, n2,
  the number of free positive and negative ions respectively per cubic
  centimetre of the gas. The number of collisions between positive and
  negative ions per second in one cubic centimetre of the gas is
  proportional to n1n2. If a certain fraction of the collisions between
  the positive and negative ions result in the formation of an
  electrically neutral system, the number of ions which disappear per
  second on a cubic centimetre will be equal to [alpha]n1 n2, where
  [alpha] is a quantity which is independent of n1, n2; hence if t is
  the time since the ionizing agent was applied to the gas, we have

    dn1/dt = q - [alpha]n1 n2, dn2/dt = q - [alpha]n1 n2.

  Thus n1 - n2 is constant, so if the gas is uncharged to begin with, n1
  will always equal n2. Putting n1 = n2 = n we have

    dn/dt = q - [alpha]n²              (1),

  the solution of which is, since n = 0 when t = 0,

      k([epsilon]^{2k[alpha]t} - 1)
  n = ----------------------------     (2)
       [epsilon]^{2k[alpha]t} + 1

  if k² = q/[alpha]. Now the number of ions when the gas has reached a
  steady state is got by putting t equal to infinity in the preceding
  equation, and is therefore given by the equation

    n0 = k = [root](q/[alpha]).

  We see from equation (1) that the gas will not approximate to its
  steady state until 2k[alpha]t is large, that is until t is large
  compared with 1/2k[alpha] or with 1/2[root](q[alpha]). We may thus
  take 1/2[root](q[alpha]) as a measure of the time taken by the gas to
  reach a steady state when exposed to an ionizing agent; as this time
  varies inversely as [root]q we see that when the ionization is feeble
  it may take a very considerable time for the gas to reach a steady
  state. Thus in the case of our atmosphere where the production of ions
  is only at the rate of about 30 per cubic centimetre per second, and
  where, as we shall see, [alpha] is about 10^-6, it would take some
  minutes for the ionization in the air to get into a steady state if
  the ionizing agent were suddenly applied.

  We may use equation (1) to determine the rate at which the ions
  disappear when the ionizing agent is removed. Putting q=0 in that
  equation we get dn/[alpha]t = -[alpha]n².

  Hence  n = n0/(1 + n0[alpha]t)       (3),

  where n0 is the number of ions when t = 0. Thus the number of ions
  falls to one-half its initial value in the time 1/n0[alpha]. The
  quantity [alpha] is called the _coefficient of recombination_, and its
  value for different gases has been determined by Rutherford (_Phil.
  Mag._ 1897 [5], 44, p. 422), Townsend (_Phil. Trans._, 1900, 193, p.
  129), McClung (_Phil. Mag._, 1902 [6], 3, p. 283), Langevin (_Ann.
  chim. phys._ [7], 28, p. 289), Retschinsky (_Ann. d. Phys._, 1905, 17,
  p. 518), Hendred (_Phys. Rev._, 1905, 21, p. 314). The values of
  [alpha]/e, e being the charge on an ion in electrostatic measure as
  determined by these observers for different gases, is given in the
  following  table:--

    +-----+----------+----------+----------+------------+----------+
    |     | Townsend.| McClung. | Langevin.|Retschinsky.| Hendred. |
    +-----+----------+----------+----------+------------+----------+
    | Air |   3420   |   3380   |   3200   |    4140    |   3500   |
    | O2  |   3380   |          |          |            |          |
    | CO2 |   3500   |   3490   |   3400   |            |          |
    | H2  |   3020   |   2940   |          |            |          |
    +-----+----------+----------+----------+------------+----------+

  The gases in these experiments were carefully dried and free from
  dust; the apparent value of [alpha] is much increased when dust or
  small drops of water are present in the gas, for then the ions get
  caught by the dust particles, the mass of a particle is so great
  compared with that of an ion that they are practically immovable under
  the action of the electric field, and so the ions clinging to them
  escape detection when electrical methods are used. Taking e as 3.5 ×
  10^-10, we see that [alpha] is about 1.2 × 10^-6, so that the number
  of recombinations in unit time between n positive and n negative ions
  in unit volume is 1.2 × 10^-6n². The kinetic theory of gases shows
  that if we have n molecules of air per cubic centimetre, the number of
  collisions per second is 1.2 × 10^-10n² at a temperature of 0° C.
  Thus we see that the number of recombinations between oppositely
  charged ions is enormously greater than the number of collisions
  between the same number of neutral molecules. We shall see that the
  difference in size between the ion and the molecule is not nearly
  sufficient to account for the difference between the collisions in the
  two cases; the difference is due to the force between the oppositely
  charged ions, which drags ions into collisions which but for this
  force would have missed each other.

  Several methods have been used to measure [alpha]. In one method air,
  exposed to some ionizing agent at one end of a long tube, is slowly
  sucked through the tube and the saturation current measured at
  different points along the tube. These currents are proportional to
  the values of n at the place of observation: if we know the distance
  of this place from the end of the tube when the gas was ionized and
  the velocity of the stream of gas, we can find t in equation (3), and
  knowing the value of n we can deduce the value of [alpha] from the
  equation

    1/n1 - 1/n2 = [alpha](t1 - t2),

  where n1, n2 are the values of n at the times t1, t2 respectively. In
  this method the tubes ought to be so wide that the loss of ions by
  diffusion to the sides of the tube is negligible. There are other
  methods which involve the knowledge of the speed with which the ions
  move under the action of known electric forces; we shall defer the
  consideration of these methods until we have discussed the question of
  these speeds.

  In measuring the value of [alpha] it should be remembered that the
  theory of the methods supposes that the ionization is uniform
  throughout the gas. If the total ionization throughout a gas remains
  constant, but instead of being uniformly distributed is concentrated
  in patches, it is evident that the ions will recombine more quickly in
  the second case than in the first, and that the value of [alpha] will
  be different in the two cases. This probably explains the large values
  of [alpha] obtained by Retschinsky, who ionized the gas by the [alpha]
  rays from radium, a method which produces very patchy ionization.

  _Variation of [alpha] with the Pressure of the Gas._--All observers
  agree that there is little variation in [alpha] with the pressures for
  pressures of between 5 and 1 atmospheres; at lower pressures, however,
  the value of [alpha] seems to diminish with the pressure: thus
  Langevin (_Ann. chim. phys._, 1903, 28, p. 287) found that at a
  pressure of 1/5 of an atmosphere the value of [alpha] was about 1/5 of
  its value at atmospheric pressure.

  _Variation of [alpha] with the Temperature._--Erikson (_Phil. Mag._,
  Aug. 1909) has shown that the value of [alpha] for air increases as
  the temperature diminishes, and that at the temperature of liquid air
  -180° C., it is more than twice as great as at +12° C.

  Since, as we have seen, the recombination is due to the coming
  together of the positive and negative ions under the influence of the
  electrical attraction between them, it follows that a large electric
  force sufficient to overcome this attraction would keep the ions apart
  and hence diminish the coefficient of recombination. Simple
  considerations, however, will show that it would require exceedingly
  strong electric fields to produce an appreciable effect. The value of
  [alpha] indicates that for two oppositely charged ions to unite they
  must come within a distance of about 1.5 × 10^-6 centimetres; at this
  distance the attraction between them is e² × 10^12/2.25, and if X is
  the external electric force, the force tending to pull them apart
  cannot be greater than Xe; if this is to be comparable with the
  attraction, X must be comparable with e × 10^12/2.25, or putting e = 4
  × 10^-10, with 1.8 × 10²; this is 54,000 volts per centimetre, a
  force which could not be applied to gas at atmospheric pressure
  without producing a spark.

  _Diffusion of the Ions._--The ionized gas acts like a mixture of
  gases, the ions corresponding to two different gases, the non-ionized
  gas to a third. If the concentration of the ions is not uniform, they
  will diffuse through the non-ionized gas in such a way as to produce a
  more uniform distribution. A very valuable series of determinations of
  the coefficient of diffusion of ions through various gases has been
  made by Townsend (_Phil. Trans._, 1900, A, 193, p. 129). The method
  used was to suck the ionized gas through narrow tubes; by measuring
  the loss of both the positive and negative ions after the gases had
  passed through a known length of tube, and allowing for the loss by
  recombination, the loss by diffusion and hence the coefficient of
  diffusion could be determined. The following tables give the values of
  the coefficients of diffusion D on the C.G.S. system of units as
  determined by  Townsend:--

    Table I.--_Coefficients of Diffusion (D) in Dry Gases._

    +-----+------------+------------+----------+------------------+
    |Gas. |D for +ions.|D for -ions.|Mean Value|  Ratio of D for  |
    |     |            |            |   of D.  | - to D for +ions.|
    +-----+------------+------------+----------+------------------+
    | Air |    .028    |    .043    |   .0347  |       1.54       |
    | O2  |    .025    |    .0396   |   .0323  |       1.58       |
    | CO2 |    .023    |    .026    |   .0245  |       1.13       |
    | H2  |    .123    |    .190    |   .156   |       1.54       |
    +-----+------------+------------+----------+------------------+

    Table II.--Coefficients of Diffusion in Moist Gases.

    +-----+------------+------------+----------+------------------+
    |Gas. |D for +ions.|D for -ions.|Mean Value|  Ratio of D for  |
    |     |            |            |   of D.  | - to D for +ions.|
    +-----+------------+------------+----------+------------------+
    | Air |    .032    |    .037    |   .0335  |       1.09       |
    | O2  |    .0288   |    .0358   |   .0323  |       1.24       |
    | CO2 |    .0245   |    .0255   |   .025   |       1.04       |
    | H2  |    .128    |    .142    |   .135   |       1.11       |
    +-----+------------+------------+----------+------------------+

  It is interesting to compare with these coefficients the values of D
  when various gases diffuse through each other. D for hydrogen through
  air is .634, for oxygen through air .177, for the vapour of isobutyl
  amide through air .042. We thus see that the velocity of diffusion of
  ions through air is much less than that of the simple gas, but that it
  is quite comparable with that of the vapours of some complex organic
  compounds.

  The preceding tables show that the negative ions diffuse more rapidly
  than the positive, especially in dry gases. The superior mobility of
  the negative ions was observed first by Zeleny (_Phil. Mag._, 1898
  [5], 46, p. 120), who showed that the velocity of the negative ions
  under an electric force is greater than that of the positive. It will
  be noticed that the difference between the mobility of the negative
  and the positive ions is much more pronounced in dry gases than in
  moist. The difference in the rates of diffusion of the positive and
  negative ions is the reason why ionized gas, in which, to begin with,
  the positive and negative charges were of equal amounts, sometimes
  becomes electrified even although the gas is not acted upon by
  electric forces. Thus, for example, if such gas be blown through
  narrow tubes, it will be positively electrified when it comes out, for
  since the negative ions diffuse more rapidly than the positive, the
  gas in its passage through the tubes will lose by diffusion more
  negative than positive ions and hence will emerge positively
  electrified. Zeleny snowed that this effect does not occur when, as in
  carbonic acid gas, the positive and negative ions diffuse at the same
  rates. Townsend (loc. cit.) showed that the coefficient of diffusion
  of the ions is the same whether the ionization is produced by Röntgen
  rays, radioactive substances, ultra-violet light, or electric sparks.
  The ions produced by chemical reactions and in flames are much less
  mobile; thus, for example, Bloch (_Ann. chim. phys._, 1905 [8], 4, p.
  25) found that for the ions produced by drawing air over phosphorus
  the value of [alpha]/e was between 1 and 6 instead of over 3000, the
  value when the air was ionized by Röntgen rays.

_Velocity of Ions in an Electric Field._--The velocity of ions in an
electric field, which is of fundamental importance in conduction, is
very closely related to the coefficient of diffusion. Measurements of
this velocity for ions produced by Röntgen rays have been made by
Rutherford (_Phil. Mag._ [5], 44, p. 422), Zeleny (_Phil. Mag._ [5], 46,
p. 120), Langevin (_Ann. Chim. Phys._, 1903, 28, p. 289), Phillips
(_Proc. Roy. Soc._ 78, A, p. 167), and Wellisch (_Phil. Trans._, 1909,
209, p. 249). The ions produced by radioactive substance have been
investigated by Rutherford (_Phil. Mag._ [5], 47, p. 109) and by Franck
and Pohl (_Verh. deutsch. phys. Gesell._, 1907, 9, p. 69), and the
negative ions produced when ultra-violet light falls on a metal plate by
Rutherford (_Proc. Camb. Phil. Soc._ 9, p. 401). H. A. Wilson (_Phil.
Trans._ 192, p. 4O9), Marx (_Ann. de Phys._ 11, p. 765), Moreau (_Journ.
de Phys._ 4, 11, p. 558; _Ann. Chim. Phys._ 7, 30, p. 5) and Gold
(_Proc. Roy. Soc._ 79, p. 43) have investigated the velocities of ions
produced by putting various salts into flames; McClelland (_Phil. Mag._
46, p. 29) the velocity of the ions in gases sucked from the
neighbourhood of flames and arcs; Townsend (_Proc. Camb. Phil. Soc._ 9,
p. 345) and Bloch (_loc. cit._) the velocity of ions produced by
chemical reaction; and Chattock (_Phil. Mag._ [5], 48, p. 401) the
velocity of the ions produced when electricity escapes from a sharp
needle point into a gas.

Several methods have been employed to determine these velocities. The
one most frequently employed is to find the electromotive intensity
required to force an ion against the stream of gas moving with a known
velocity parallel to the lines of electric force. Thus, of two
perforated plane electrodes vertically over each other, suppose the
lower to be positively, the upper negatively electrified, and suppose
that the gas is streaming vertically downwards with the velocity V; then
unless the upward velocity of the positive ion is greater than V, no
positive electricity will reach the upper plate. If we increase the
strength of the field between the plates, and hence the upward velocity
of the positive ion, until the positive ions just begin to reach the
upper plate, we know that with this strength of field the velocity of
the positive ion is equal to V. By this method, which has been used by
Rutherford, Zeleny and H. A. Wilson, the velocity of ions in fields of
various strengths has been determined.

  The arrangement used by Zeleny is represented in fig. 8. P and Q are
  square brass plates. They are bored through their centres, and to the
  openings the tubes R and S are attached, the space between the plates
  being covered in so as to form a closed box. K is a piece of wire
  gauze completely covering the opening in Q; T is an insulated piece of
  wire gauze nearly but not quite filling the opening in the plate P,
  and connected with one pair of quadrants of an electrometer E. A plug
  of glass wool G filters out the dust from a stream of gas which enters
  the vessel by the tube D and leaves it by F; this plug also makes the
  velocity of the flow of the gas uniform across the section of the
  tube. The Röntgen rays to ionize the gas were produced by a bulb at
  O, the bulb and coil being in a lead-covered box, with an aluminium
  window through which the rays passed. Q is connected with one pole of
  a battery of cells, P and the other pole of the battery are put to
  earth. The changes in the potential of T are due to ions giving up
  their charges to it. With a given velocity of air-blast the potential
  of T was found not to change unless the difference of potential
  between P and Q exceeded a critical value. The field corresponding to
  this critical value thus made the ions move with the known velocity of
  the blast.

  [Illustration: FIG. 8.]

  [Illustration: FIG. 9.]

  Another method which has been employed by Rutherford and McClelland is
  based on the action of an electric field in destroying the
  conductivity of gas streaming through it. Suppose that BAB, DCD (fig.
  9) are a system of parallel plates boxed in so that a stream of gas,
  after flowing between BB, passes between DD without any loss of gas in
  the interval. Suppose the plates DD are insulated, and connected with
  one pair of quadrants of an electrometer, by charging up C to a
  sufficiently high potential we can drive all the positive ions which
  enter the system DCD against the plates D; this will cause a deflexion
  of the electrometer, which in one second will be proportional to the
  number of positive ions which have entered the system in that time. If
  we charge A up to a high potential, B being put to earth, we shall
  find that the deflexion of the electrometer connected with DD is less
  than it was when A and B were at the same potential, because some of
  the positive ions in their passage through BAB are driven against the
  plates B. If u is the velocity along the lines of force in the uniform
  electric field between A and B, and t the time it takes for the gas to
  pass through BAB, then all the positive ions within a distance ut of
  the plates B will be driven up against these plates, and thus if the
  positive ions are equally distributed through the gas, the number of
  positive ions which emerge from the system when the electric field is
  on will bear to the number which emerge when the field is off the
  ratio of 1 - ut/l to unity, where l is the distance between A and B.
  This ratio is equal to the ratio of the deflexions in one second of
  the electrometer attached to D, hence the observations of this
  instrument give 1 - ut/l. If we know the velocity of the gas and the
  length of the plates A and B, we can determine t, and since l can be
  easily measured, we can find u, the velocity of the positive ion in a
  field of given strength. By charging A and C negatively instead of
  positively we can arrive at the velocity of the negative ion. In
  practice it is more convenient to use cylindrical tubes with coaxial
  wires instead of the systems of parallel plates, though in this case
  the calculation of the velocity of the ions from the observations is a
  little more complicated, inasmuch as the electric field is not uniform
  between the tubes.

  [Illustration: FIG. 10.]

  A method which gives very accurate results, though it is only
  applicable in certain cases, is the one used by Rutherford to measure
  the velocity of the negative ions produced close to a metal plate by
  the incidence on the plate of ultra-violet light. The principle of the
  method is as follows:--AB (fig. 10) is an insulated horizontal plate
  of well-polished zinc, which can be moved vertically up and down by
  means of a screw; it is connected with one pair of quadrants of an
  electrometer, the other pair of quadrants being put to earth. CD is a
  base-plate with a hole EF in it; this hole is covered with fine wire
  gauze, through which ultra-violet light passes and falls on the plate
  AB. The plate CD is connected with an alternating current dynamo,
  which produces a simply-periodic potential difference between AB and
  CD, the other pole being put to earth. Suppose that at any instant the
  plate CD is at a higher potential than AB, then the negative ions from
  AB will move towards CD, and will continue to do so as long as the
  potential of CD is higher than that of AB. If, however, the potential
  difference changes sign before the negative ions reach CD, these ions
  will go back to AB. Thus AB will not lose any negative charge unless
  the distance between the plates AB and CD is less than the distance
  traversed by the negative ion during the time the potential of CD is
  higher than that of AB. By altering the distance between the plates
  until CD just begins to lose a negative charge, we find the velocity
  of the negative ion under unit electromotive intensity. For suppose
  the difference of potential between AB and CD is equal to a sin pt,
  then if d is the distance between the plates, the electric intensity
  is equal to a sin pt/d; if we suppose the velocity of the ion is
  proportional to the electric intensity, and if u is the velocity for
  unit electric intensity, the velocity of the negative ion will be ua
  sin pt/d. Hence if x represent the distance of the ion from AB

    dx    ua
    --- = --- sin pt
    dT    d

         ua
    x = ----(1 - cos pt), if x = 0 when t = 0.
         pd

  Thus the greatest distance the ion can get from the plate is equal to
  2au/pd, and if the distance between the plates is gradually reduced to
  this value, the plate AB will begin to lose a negative charge; hence
  when this happens

    d = 2au/pd, or u = pd²/2a,

  an equation by means of which we can find u.

  In this form the method is not applicable when ions of both signs are
  present. Franck and Pohl (_Verh. deutsch. physik. Gesell._ 1907, 9, p.
  69) have by a slight modification removed this restriction. The
  modification consists in confining the ionization to a layer of gas
  below the gauze EF. If the velocity of the positive ions is to be
  determined, these ions are forced through the gauze by applying to the
  ionized gas a small constant electric force acting upwards; if
  negative ions are required, the constant force is reversed. After
  passing through the gauze the ions are acted upon by alternating
  forces as in Rutherford's method.

  Langevin (_Ann. chim. phys._, 1903, 28, p. 289) devised a method of
  measuring the velocity of the ions which has been extensively used; it
  has the advantage of not requiring the rate of ionization to remain
  uniform. The general idea is as follows. Suppose that we expose the
  gas between two parallel plates A, B to Röntgen rays or some other
  ionizing agent, then stop the rays and apply a uniform electric field
  to the region between the plates. If the force on the positive ion is
  from A to B, the plate B will receive a positive charge of
  electricity. After the electric force has acted for a time T reverse
  it. B will now begin to receive negative electricity and will go on
  doing so until the supply of negative ions is exhausted. Let us
  consider how the quantity of positive electricity received by B will
  vary with T. To fix our ideas, suppose the positive ions move more
  slowly than the negative; let T2 and T1 be respectively the times
  taken by the positive and negative ions to move under the electric
  field through a distance equal to AB, the distance between the planes.
  Then if T is greater than T2 all the ions will have been driven from
  between the plates before the field is reversed, and therefore the
  positive charge received by B will not depend upon T. Next let T be
  less than T2 but greater than T1; then at the time when the field is
  reversed all the negative ions will have been driven from between the
  plates, so that the positive charge received by B will not be
  neutralized by the arrival of fresh ions coming to it after the
  reversal of the field. The number of positive ions driven against the
  plate B will be proportional to T. Thus if we measure the value of the
  positive charge on B for a series of values of T, each value being
  less than the preceding, we shall find that until T reaches a certain
  value the charge remains constant, but as soon as we reduce the time
  below this value the charge diminishes. The value of T when the
  diminution in the field begins is T2, the time taken for a positive
  ion to cross from A to B under the electric field; thus from T2 we can
  calculate the velocity of the positive ion in this field. If we still
  further diminish T, we shall find that we reach a value when the
  diminution of the positive charge on B with the time suddenly becomes
  much more rapid; this change occurs when T falls below T1 the time
  taken for the negative ions to go from one plate to the other, for now
  when the field is reversed there are still some negative ions left
  between the plates, and these will be driven against B and rob it of
  some of the positive charge it had acquired before the field was
  reversed. By observing the time when the increase in the rate of
  diminution of the positive charge with the time suddenly sets in we
  can determine T1, and hence the velocity of the negative ions.

  The velocity of the ions produced by the discharge of electricity from
  a fine point was determined by Chattock by an entirely different
  method. In this case the electric field is so strong and the velocity
  of the ion so great that the preceding methods are not applicable.
  Suppose P represents a vertical needle discharging electricity into
  air, consider the force acting on the ions included between two
  horizontal planes A, B. If P is the density of the electrification,
  and Z the vertical component of the electric intensity, F the
  resultant force on the ions between A and B is vertical and equal to
      _ _ _
     / / /
     | | | Z[rho]dxdydz.
    _/_/_/

  Let us suppose that the velocity of the ion is proportional to the
  electric intensity, so that if w is the vertical velocity of the ions,
  which are supposed all to be of one sign, w = RZ.

  Substituting this value of Z, the vertical force on the ions between A
  and B is equal to
        _ _ _
    1  / / /
    -  | | | w[rho]dxdydz.
    R _/_/_/

  But [integral][integral]w[rho]dxdy = [iota], where [iota] is the
  current streaming from the point. This current, which can be easily
  measured by putting a galvanometer in series with the discharging
  point, is independent of z, the vertical distance of a plane between A
  and B below the charging point. Hence we have
                _
        [iota] /       [iota]
    F = ------ | dz  = ------·z.
          R   _/         R

  This force must be counterbalanced by the difference of gaseous
  pressures over the planes A and B; hence if pB and pA denote
  respectively the pressures over B and A, we have

              [iota]
    pB - pA = ------ z.
                R

  Hence by the measurement of these pressures we can determine R, and
  hence the velocity with which an ion moves under a given electric
  intensity.

  There are other methods of determining the velocities of the ions, but
  as these depend on the theory of the conduction of electricity through
  a gas containing charged ions, we shall consider them in our
  discussion of that theory.

  By the use of these methods it has been shown that the velocities of
  the ions in a given gas are the same whether the ionization is
  produced by Röntgen rays, radioactive substances, ultra-violet light,
  or by the discharge of electricity from points. When the ionization is
  produced by chemical action the ions are very much less mobile, moving
  in the same electric field with a velocity less than one-thousandth
  part of the velocity of the first kind of ions. On the other hand, as
  we shall see later, the velocity of the negative ions in flames is
  enormously greater than that of even the first kind of ion under
  similar electric fields and at the same pressure. But when these
  negative ions get into the cold part of the flame, they move
  sluggishly with velocities of the order of those possessed by the
  second kind. The results of the various determinations of the
  velocities of the ions are given in the following table. The
  velocities are in centimetres per second under an electric force of
  one volt per centimetre, the pressure of the gas being 1 atmosphere.
  V+ denotes the velocity of the positive ion, V- that of the negative.
  V is the mean velocity of the positive and negative ions.

    _Velocities of Ions.--Ions produced by Röntgen Rays._

    +----------------------+------+------+------+-----------------+
    |         Gas.         |  V+. |  V-. |  V.  |    Observer.    |
    +----------------------+------+------+------+-----------------+
    | Air                  |  ..  |  ..  | 1.6  | Rutherford      |
    | Air (dry)            | 1.36 | 1.87 |  ..  | Zeleny          |
    |    "                 | 1.60 | 1.70 |  ..  | Langevin        |
    |    "                 | 1.39 | 1.78 |  ..  | Phillips        |
    |    "                 | 1.54 | 1.78 |  ..  | Wellisch        |
    | Air (moist)          | 1.37 | 1.81 |  ..  | Zeleny          |
    | Oxygen (dry)         | 1.36 | 1.80 |  ..  |    "            |
    | Oxygen (moist)       | 1.29 | 1.52 |  ..  |    "            |
    | Carbonic acid (dry)  | 0.76 | 0.81 |  ..  |    "            |
    |        "             | 0.86 | 0.90 |  ..  | Langevin        |
    |        "             | 0.81 | 0.85 |  ..  | Wellisch        |
    | Carbonic acid (moist)| 0.82 | 0.75 |  ..  | Zeleny          |
    | Hydrogen (dry)       | 6.70 | 7.95 |  ..  |    "            |
    | Nitrogen             |  ..  |  ..  | 1.6  | Rutherford      |
    | Sulphur dioxide      | 0.44 | 0.41 |  ..  | Wellisch        |
    | Hydrochloric acid    |  ..  |  ..  | 1.27 | Rutherford      |
    | Chlorine             |  ..  |  ..  | 1.0  |    "            |
    | Helium (dry)         | 5.09 | 6.31 |  ..  | Franck and Pohl |
    | Carbon monoxide      | 1.10 | 1.14 |  ..  | Wellisch        |
    | Nitrous oxide        | 0.82 | 0.90 |  ..  |    "            |
    | Ammonia              | 0.74 | 0.80 |  ..  |    "            |
    | Aldehyde             | 0.31 | 0.30 |  ..  |    "            |
    | Ethyl alcohol        | 0.34 | 0.27 |  ..  |    "            |
    | Acetone              | 0.31 | 0.29 |  ..  |    "            |
    | Ethyl chloride       | 0.33 | 0.31 |  ..  |    "            |
    | Pentane              | 0.36 | 0.35 |  ..  |    "            |
    | Methyl acetate       | 0.33 | 0.36 |  ..  |    "            |
    | Ethyl formate        | 0.30 | 0.31 |  ..  |    "            |
    | Ethyl ether          | 0.29 | 0.31 |  ..  |    "            |
    | Ethyl acetate        | 0.31 | 0.28 |  ..  |    "            |
    | Methyl bromide       | 0.29 | 0.28 |  ..  |    "            |
    | Methyl iodide        | 0.21 | 0.22 |  ..  |    "            |
    | Carbon tetrachloride | 0.30 | 0.31 |  ..  |    "            |
    | Ethyl iodide         | 0.17 | 0.16 |  ..  |    "            |
    +----------------------+------+------+------+-----------------+

          _Ions produced by Ultra-Violet Light._

    Air                          1.4           Rutherford
    Hydrogen                     3.9           Rutherford
    Carbonic acid                0.78          Rutherford

          _Ions in Gases sucked from Flames._

    Velocities varying from .04 to .23         McClelland

          _Ions in Flames containing Salts._

    Negative ions               12.9 cm./sec.  Gold
    +ions for salts of Li, Na,
           K, Rb, Cs            62             H. A. Wilson
             "                 200             Marx
             "                  80             Moreau

         _Ions liberated by Chemical Action._

    Velocities of the order of 0.0005 cm./sec. Bloch

    _Ions from Point Discharge._

    +---------------+------+-------+------+----------+
    | Hydrogen      | 5.4  | 7.43  | 6.41 | Chattock |
    | Carbonic acid | 0.83 | 0.925 | 0.88 | Chattock |
    | Air           | 1.32 | 1.80  | 1.55 | Chattock |
    | Oxygen        | 1.30 | 1.85  | 1.57 | Chattock |
    +---------------+------+-------+------+----------+

  It will be seen from this table that the greater mobility of the
  negative ions is very much more marked in the case of the lighter and
  simpler gases than in that of the heavier and more complicated ones;
  with the vapours of organic substances there seems but little
  difference between the mobilities of the positive and negative ions,
  indeed in one or two cases the positive one seems slightly but very
  slightly the more mobile of the two. In the case of the simple gases
  the difference is much greater when the gases are dry than when they
  are moist. It has been shown by direct experiment that the velocities
  are directly proportional to the electric force.

  _Variation of Velocities with Pressure._--Until the pressure gets low
  the velocities of the ions, negative as well as positive, vary
  inversely as the pressure. Langevin (loc. cit.) was the first to show
  that at very low pressures the velocity of the negative ions increases
  more rapidly as the pressure is diminished than this law indicates. If
  the nature of the ion did not change with the pressure, the kinetic
  theory of gases indicates that the velocity would vary inversely as
  the pressure, so that Langevin's results indicate a change in the
  nature of the negative ion when the pressure is diminished below a
  certain value. Langevin's results are given in the following table,
  where p represents the pressure measured in centimetres of mercury, V+
  and V- the velocities of the positive and negative ions in air under
  unit electrostatic force, i.e. 300 volts per centimetre:--

    +----------------------+----------------------+
    |     Negative Ions.   |    Positive Ions.    |
    +-------+------+-------+-------+------+-------+
    |   p.  |  V-. |pV-/76.|   p.  |  V+. |pV+/76.|
    +-------+------+-------+-------+------+-------+
    |   7.5 | 6560 |  647  |   7.5 | 4430 |  437  |
    |  20.0 | 2204 |  580  |  20.0 | 1634 |  430  |
    |  41.5 |  994 |  530  |  41.5 |  782 |  427  |
    |  76.0 |  510 |  510  |  76.0 |  480 |  420  |
    | 142.0 |  270 |  505  | 142.0 |  225 |  425  |
    +-------+------+-------+-------+------+-------+

  The increase in the case of pV- indicates that the structure of the
  negative ion gets simpler as the pressure is reduced. Wallisch in some
  experiments made at the Cavendish Laboratory found that the diminution
  in the value of pV- at low pressures is much more marked in some gases
  than in others, and in some gases he failed to detect it; but it must
  be remembered that it is difficult to get measurements at pressures of
  only a few millimetres, as the amount of ionization is so exceedingly
  small at such pressures that the quantities to be observed are hardly
  large enough to admit of accurate measurements by the methods
  available at higher pressures.

  _Effect of Temperature on the Velocity of the Ions._--Phillips (_Proc.
  Roy. Soc._, 1906, 78, p. 167) investigated, using Langevin's method,
  the velocities of the + and - ions through air at atmospheric pressure
  at temperatures ranging from that of boiling liquid air to 411° C.; R1
  and R2 are the velocities of the + and - ions respectively when the
  force is a volt per centimetre.

    +-------+-------+---------------------+
    |  R1.  |  R2.  |Temperature Absolute.|
    +-------+-------+---------------------+
    | 2.00  | 2.495 |        411°         |
    | 1.95  | 2.40  |        399°         |
    | 1.85  | 2.30  |        383°         |
    | 1.81  | 2.21  |        373°         |
    | 1.67  | 2.125 |        348°         |
    | 1.60  | 2.00  |        333°         |
    | 1.39  | 1.785 |        285°         |
    | 0.945 | 1.23  |        209°         |
    | 0.235 | 0.235 |        94°          |
    +-------+-------+---------------------+

  We see that except in the case of the lowest temperature, that of
  liquid air, where there is a great drop in the velocity, the
  velocities of the ions are proportional to the absolute temperature.
  On the hypothesis of an ion of constant size we should, from the
  kinetic theory of gases, expect the velocity to be proportional to the
  square root of the absolute temperature, if the charge on the ion did
  not affect the number of collisions between the ion and the molecules
  of the gas through which it is moving. If the collisions were brought
  about by the electrical attraction between the ions and the molecules,
  the velocity would be proportional to the absolute temperature. H. A.
  Wilson (_Phil. Trans._ 192, p. 499), in his experiments on the
  conduction of flames and hot gases into which salts had been put,
  found that the velocity of the positive ions in flames at a
  temperature of 2000° C. containing the salts of the alkali metals was
  62 cm./sec. under an electric force of one volt per centimetre, while
  the velocity of the positive ions in a stream of hot air at 1000° C.
  containing the same salts was only 7 cm./sec. under the same force.
  The great effect of temperature is also shown in some experiments of
  McClelland (_Phil. Mag._ [5], 46, p. 29) on the velocities of the ions
  in gases drawn from Bunsen flames and arcs; he found that these
  depended upon the distance the gas had travelled from the flame. Thus,
  the velocity of the ions at a distance of 5.5 cm. from the Bunsen
  flame when the temperature was 230° C. was .23 cm./sec. for a volt per
  centimetre; at a distance of 10 cm. from the flame when the
  temperature was 160° C. the velocity was .21 cm./sec; while at a
  distance of 14.5 cm. from the flame when the temperature was 105° C.
  the velocity was only .04 cm./sec. If the temperature of the gas at
  this distance from the flame was raised by external means, the
  velocity of the ions increased.

  We can derive some information as to the constitution of the ions by
  calculating the velocity with which a molecule of the gas would move
  in the electric field if it carried the same charge as the ion. From
  the theory of the diffusion of gases, as developed by Maxwell, we know
  that if the particles of a gas A are surrounded by a gas B, then, if
  the partial pressure of A is small, the velocity u with which its
  particles will move when acted upon by a force Xe is given by the
  equation

           Xe
    u = ------- D,
        (p1/N1)

  where D represents the coefficient of inter-diffusion of A into B, and
  N1 the number of particles of A per cubic centimetre when the pressure
  due to A is p1. Let us calculate by this equation the velocity with
  which a molecule of hydrogen would move through hydrogen if it carried
  the charge carried by an ion, which we shall prove shortly to be equal
  to the charge carried by an atom of hydrogen in the electrolysis of
  solutions. Since p1/N1 is independent of the pressure, it is equal to
  [Pi]/N, where [Pi] is the atmospheric pressure and N the number of
  molecules in a cubic centimetre of gas at atmospheric pressure. Now Ne
  = 1.22 × 10^10, if e is measured in electrostatic units; [Pi] = 10^6
  and D in this case is the coefficient of diffusion of hydrogen into
  itself, and is equal to 1.7. Substituting these values we find

    u = 1.97 × 10^4X.

  If the potential gradient is 1 volt per centimetre, X = 1/300.
  Substituting this value for X, we find u = 66 cm./sec, for the
  velocity of a hydrogen molecule. We have seen that the velocity of the
  ion in hydrogen is only about 5 cm./sec, so that the ion moves more
  slowly than it would if it were a single molecule. One way of
  explaining this is to suppose that the ion is bigger than the
  molecule, and is in fact an aggregation of molecules, the charged ion
  acting as a nucleus around which molecules collect like dust round a
  charged body. This view is supported by the effect produced by
  moisture in diminishing the velocity of the negative ion, for, as C.
  T. R. Wilson (_Phil. Trans._ 193, p. 289) has shown, moisture tends to
  collect round the ions, and condenses more easily on the negative than
  on the positive ion. In connexion with the velocities of ions in the
  gases drawn from flames, we find other instances which suggest that
  condensation takes place round the ions. An increase in the size of
  the system is not, however, the only way by which the velocity might
  fall below that calculated for the hydrogen molecule, for we must
  remember that the hydrogen molecule, whose coefficient of diffusion is
  1.7, is not charged, while the ion is. The forces exerted by the ion
  on the other molecules of hydrogen are not the same as those which
  would be exerted by a molecule of hydrogen, and as the coefficient of
  diffusion depends on the forces between the molecules, the coefficient
  of diffusion of a charged molecule into hydrogen might be very
  different from that of an uncharged one.

  Wellisch (_loc. cit._) has shown that the effect of the charge on the
  ion is sufficient in many cases to explain the small velocity of the
  ions, even if there were no aggregation.

  _Mixture of Gases._--The ionization of a mixture of gases raises some
  very interesting questions. If we ionize a mixture of two very
  different gases, say hydrogen and carbonic acid, and investigate the
  nature of the ions by measuring their velocities, the question arises,
  shall we find two kinds of positive and two kinds of negative ions
  moving with different velocities, as we should do if some of the
  positive ions were positively charged hydrogen molecules, while others
  were positively charged molecules of carbonic acid; or shall we find
  only one velocity for the positive ions and one for the negative? Many
  experiments have been made on the velocity of ions in mixtures of two
  gases, but as yet no evidence has been found of the existence of two
  different kinds of either positive or negative ions in such mixtures,
  although some of the methods for determining the velocities of the
  ions, especially Langevin's, ought to give evidence of this effect, if
  it existed. The experiments seem to show that the positive (and the
  same is true for the negative) ions in a mixture of gases are all of
  the same kind. This conclusion is one of considerable importance, as
  it would not be true if the ions consisted of single molecules of the
  gas from which they are produced.

  _Recombination._--Several methods enable us to deduce the coefficient
  of recombination of the ions when we know their velocities. Perhaps
  the simplest of these consists in determining the relation between the
  current passing between two parallel plates immersed in ionized gas
  and the potential difference between the plates. For let q be the
  amount of ionization, i.e. the number of ions produced per second per
  unit volume of the gas, A the area of one of the plates, and d the
  distance between them; then if the ionization is constant through the
  volume, the number of ions of one sign produced per second in the gas
  is qAd. Now if i is the current per unit area of the plate, e the
  charge on an ion, iA/e ions of each sign are driven out of the gas by
  the current per second. In addition to this source of loss of ions
  there is the loss due to the recombination; if n is the number of
  positive or negative ions per unit volume, then the number which
  recombine per second is [alpha]n² per cubic centimetre, and if n is
  constant through the volume of the gas, as will approximately be the
  case if the current through the gas is only a small fraction of the
  saturation current, the number of ions which disappear per second
  through recombination is [alpha]n²·Ad. Hence, since when the gas is
  in a steady state the number of ions produced must be equal to the
  number which disappear, we have

    qAd = iA/e + [alpha]n²·Ad,
      q = i/ed + [alpha]^n2.

  If u1 and u2 are the velocities with which the positive and negative
  ions move, nu1e and nu2e are respectively the quantities of positive
  electricity passing in one direction through unit area of the gas per
  second, and of negative in the opposite direction, hence

    i = nu1e + nu2e.

  If X is the electric force acting on the gas, k1 and k2 the velocities
  of the positive and negative ions under unit force, u1 = k1X, u2 =
  k2X; hence

    n = i/(k1 + k2)Xe,

  and we have

        i       [alpha]i²
    q = -- + --------------.
        ed   (k1 + k2)²e²X²

  But qed is the saturation current per unit area of the plate; calling
  this I, we have

              d[alpha]i²
    I - i = -------------
            e(k1 + k2)²X²

  or

            i²·d[alpha]
    X² = ------------------.
         e(I - i)(k1 + k2)²

  Hence if we determine corresponding values of X and i we can deduce
  the value of [alpha]/e if we also know (k1 + k2). The value of I is
  easily determined, as it is the current when X is very large. The
  preceding result only applies when i is small compared with I, as it
  is only in this case that the values of n and X are uniform throughout
  the volume of the gas. Another method which answers the same purpose
  is due to Langevin (_Ann. Chim. Phys._, 1903, 28, p. 289); it is as
  follows. Let A and B be two parallel planes immersed in a gas, and let
  a slab of the gas bounded by the planes a, b parallel to A and B be
  ionized by an instantaneous flash of Röntgen rays. If A and B are at
  different electric potentials, then all the positive ions produced by
  the rays will be attracted by the negative plate and all the negative
  ions by the positive, if the electric field were exceedingly large
  they would reach these plates before they had time to recombine, so
  that each plate would receive N0 ions if the flash of Röntgen rays
  produced N0 positive and N0 negative ions. With weaker fields the
  number of ions received by the plates will be less as some of them
  will recombine before they can reach the plates. We can find the
  number of ions which reach the plates in this case in the following
  way:--In consequence of the movement of the ions the slab of ionized
  gas will broaden out and will consist of three portions, one in which
  there are nothing but positive ions,--this is on the side of the
  negative plate,--another on the side of the positive plate in which
  there are nothing but negative ions, and a portion between these in
  which there are both positive and negative ions; it is in this layer
  that recombination takes place, and here if n is the number of
  positive or negative ions at the time t after the flash of Röntgen
  rays,

    n = n0/(1 + [alpha]n0t).

  With the same notation as before, the breadth of either of the outer
  layers will in time dt increase by X(k1 + k2)dt, and the number of ions
  in it by X(k1 + k2)ndt; these ions will reach the plate, the outer
  layers will receive fresh ions until the middle one disappears, which
  it will do after a time l/X(k1 + k2), where l is the thickness of the
  slab ab of ionized gas; hence N, the number of ions reaching either
  plate, is given by the equation
         _
        / l/X(k1+k2)   n0X(k1 + k2)      X(k1 + k2)     /    n0[alpha]l \
    N = |             --------------dt = ---------- log( 1 + ----------  ).
       _/ 0           1 + n0[alpha]t      [alpha]       \    X(k1 + k2) /


  If Q is the charge received by the plate,

                   X             /    Q0[epsilon]\
    Q = Ne = -------------- log ( 1 + ----------- ),
             4[pi][epsilon]      \      4[pi]X   /

  where Q0 = n0le is the charge received by the plate when the electric
  force is large enough to prevent recombination, and [epsilon] =
  [alpha]4[pi]e(R1 + R2). We can from this result deduce the value of
  [epsilon] and hence the value of [alpha] when R1+R2 is known.

  _Distribution of Electric Force when a Current is passing through an
  Ionized Gas._--Let the two plates be at right angles to the axis of x;
  then we may suppose that between the plates the electric intensity X
  is everywhere parallel to the axis of x. The velocities of both the
  positive and negative ions are assumed to be proportional to X. Let
  k1X, k2X represent these velocities respectively; let n1, n2 be
  respectively the number of positive and negative ions per unit volume
  at a point fixed by the co-ordinate x; let q be the number of positive
  or negative ions produced in unit time per unit volume at this point;
  and let the number of ions which recombine in unit volume in unit time
  be [alpha]n1n2; then if e is the charge on the ion, the volume density
  of the electrification is (n1 - n2)e, hence

    dX
    -- = 4[pi](n1 - n2)e                 (1).
    dx

  If I is the current through unit area of the gas and if we neglect any
  diffusion except that caused by the electric field,

    n1ek1X + n2ek2X = I                  (2).

  From equations (1) and (2) we have

             1     / I     k2  dX \
    n1e = ------- (  - + ----- --  )     (3),
          k1 + k2  \ X   4[pi] dx /

             1     / I     k1  dX \
    n2e = ------- (  - - ----- --  )     (4),
          k1 + k2  \ X   4[pi] dx /

  and from these equations we can, if we know the distribution of
  electric intensity between the plates, calculate the number of
  positive and negative ions.

  In a steady state the number of positive and negative ions in unit
  volume at a given place remains constant, hence neglecting the loss by
  diffusion, we have

    d
    --(k1n1X) = q - [alpha]n1n2         (5).
    dx

    d
  - --(k2n2X) = q - [alpha]n1n2         (6).
    dx

  If k1 and K2 are constant, we have from (1), (5) and (6)

    d²X²                           / 1     1  \
    ---- = 8[pi]e(q - [alpha]n1n2)( --- + ---  )   (7),
    dx²                            \ k1    k2 /

  an equation which is very useful, because it enables us, if we know
  the distribution of X², to find whether at any point in the gas the
  ionization is greater or less than the recombination of the ions. We
  see that q - [alpha]n1n2, which is the excess of ionization over
  recombination, is proportional to d²X²/dx². Thus when the ionization
  exceeds the recombination, i.e. when q - [alpha]n1n2 is positive, the
  curve for X² is convex to the axis of x, while when the recombination
  exceeds the ionization the curve for X² will be concave to the axis of
  x. Thus, for example, fig. 11 represents the curve for X² observed by
  Graham (_Wied. Ann._ 64, p. 49) in a tube through which a steady
  current is passing. Interpreting it by equation (7), we infer that
  ionization was much in excess of recombination at A and B, slightly so
  along C, while along D the recombination exceeded the ionization.
  Substituting in equation (7) the values of n1, n2 given in (3), (4),
  we get
                   _                                                  _
    d²X²          |       [alpha]      /     k²  dX²\   /    k2   dX²\ |  / 1     1 \
    ---- = 8[pi]e |q - -------------- (1 + ----- --- ) (1 - ----- --- )| ( --- + --- )  (8).
    dx²           |_   e²X²(k1 + k2)²  \   8[pi] dx /   \   8[pi] dx /_|  \k1    k2 /

  [Illustration: Fig. 11.]

  This equation can be solved (see Thomson, _Phil. Mag._ xlvii. P. 253),
  when q is constant and k1 = k2. From the solution it appears that if
  X1 be the value of x close to one of the plates, and X0 the value
  midway between them,

                     1
    X1/X0 = ------------------
            [beta]² - 2/[beta]

  where [beta] = 8[pi]ek1/[alpha].


  Since e = 4 × 10^-10, [alpha] = 2 × 10^-6, and k1 for air at
  atmospheric pressure = 450, [beta] is about 2.3 for air at atmospheric
  pressure and it becomes much greater at lower pressures.

  Thus X1/X0 is always greater than unity, and the value of the ratio
  increases from unity to infinity as [beta] increases from zero to
  infinity. As [beta] does not involve either q or I, the ratio of X1 to
  X0 is independent of the strength of the current and of the intensity
  of the ionization.

  No general solution of equation (8) has been found when k1 is not
  equal to k2, but we can get an approximation to the solution when q is
  constant. The equations (1), (2), (3), (4) are satisfied by the
  values--

    n1 = n2 = (q / [alpha])^½

                k1
    k1n1Xe = ------- I,
             k1 + k2

                k2
    k2n2Xe = ------- I,
             k1 + k2


         /[alpha]\^½     I
    X = ( ------- )  ----------.
         \   q   /   e(k1 + k2)

  These solutions cannot, however, hold right up to the surface of the
  plates, for across each unit of area, at a point P, k1I/(k1+k2)e
  positive ions pass in unit time, and these must all come from the
  region between P and the positive plate. If [lambda] is the distance
  of P from this plate, this region cannot furnish more than q[lambda]
  positive ions, and only this number if there are no recombinations.
  Hence the solution cannot hold when q[lambda] is less than k1I/(k1 +
  k2)e, or where [lambda] is less than k1I/(k1 + k2)qe.

  Similarly the solution cannot hold nearer to the negative plate than
  the distance k2I/(k1 + k2)qe.

  [Illustration: FIG. 12.]

  The force in these layers will be greater than that in the middle of
  the gas, and so the loss of ions by recombination will be smaller in
  comparison with the loss due to the removal of the ions by the
  current. If we assume that in these layers the loss of ions by
  recombination can be neglected, we can by the method of the next
  article find an expression for the value of the electric force at any
  point in the layer. This, in conjunction with the value

          /[alpha]\^½     I
    X0 = ( ------- )  ----------
          \   q   /   e(k1 + k2)

  for the gas outside the layer, will give the value of X at any point
  between the plates. It follows from this investigation that if X1 and
  X2 are the values of X at the positive and negative plates
  respectively, and X0 the value of X outside the layer,

             /    k1     I    \^½             /    k2     I    \^½
    X1 = X0 ( I + -- --------- ) ,   X2 = X0 ( I + -- --------- ) ,
             \    k2 [epsilon]/               \    k1 [epsilon]/

  where [epsilon] = [alpha]/4[pi]e(k1 + k2). Langevin found that for air
  at a pressure of 152 mm. [epsilon] = 0.01, at 375 mm. [epsilon] =
  0.06, and at 760 mm. [epsilon] = 0.27. Thus at fairly low pressures
  1/[epsilon] is large, and we have approximately

             /k1\^½        I                   /k2\^½       I
    X1 = X0 ( -- )  ---------------,  X2 = X0 ( -- )  ---------------.
             \k2/   [root][epsilon]            \k1/   [root][epsilon]

  Therefore   X1/X2 = k1/k2,

  or the force at the positive plate is to that at the negative plate as
  the velocity of the positive ion is to that of the negative ion. Thus
  the force at the negative plate is greater than that at the positive.
  The falls of potential V1, V2 at the two layers when 1/[epsilon] is
  large can be shown to be given by the equations

                /[epsilon]\^3/2    /k1\^1/2
    V1 = 8[pi]²( --------- )   k1 ( -- )   i²,
                \q [alpha]/        \k2/

                /[epsilon]\^3/2    /k2\^1/2
    V2 = 8[pi]²( --------- )   k2 ( -- )   i²,
                \q [alpha]/        \k1/

  hence V1/V2 = k1²/k2²,

  so that the potential falls at the electrodes are proportional to the
  squares of the velocities of the ions. The change in potential across
  the layers is proportional to the square of the current, while the
  potential change between the layers is proportional to the current,
  the total potential difference between the plates is the sum of these
  changes, hence the relation between V and i will be of the form

    V = Ai + Bi².

  Mie (_Ann. der. Phys._, 1904, 13, P. 857) has by the method of
  successive approximations obtained solutions of equation (8) (i.) when
  the current is only a small fraction of the saturation current, (ii.)
  when the current is nearly saturated. The results of his
  investigations are represented in fig. 12, which represents the
  distribution of electric force along the path of the current for
  various values of the current expressed as fractions of the saturation
  current. It will be seen that until the current amounts to about
  one-fifth of the maximum current, the type of solution is the one just
  indicated, i.e. the electric force is constant except in the
  neighbourhood of the electrodes when it increases rapidly.

  Though we are unable to obtain a general solution of the equation (8),
  there are some very important special cases in which that equation can
  be solved without difficulty. We shall consider two of these, the
  first being that when the current is saturated. In this case there is
  no loss of ions by recombination, so that using the same notation as
  before we have

    d
    --(n1k1X) = q,
    dx

    d
    --(n2k2X) = -q.
    dx

  The solutions of which if q is constant are

    n1k1X = qx,

    n2k2X = I/e - qx = q(l - x),

  if l is the distance between the plates, and x = 0 at the positive
  electrode. Since

    dX/dx = 4[pi](n1 - n2)e,

  we get

      1   dX²       / 1    1 \       l
    ----- --- = qx ( -- + --  ) - q --,
    8[pi] d²x       \k1   k2 /      k2

  or

      X²      x²  / 1    1 \      lx
    ----- = q -- ( -- + --  ) - q -- + C,
    8[pi]     2   \k1   k2 /      k2

  where C is a quantity to be determined by the condition that

      _
     / l
     |  Xdx = V,
    _/0


  where V is the given potential difference between the plates. When the
  force is a minimum dX/dx = 0, hence at this point

           lk1               lk2
    x =  -------,  l - x = -------.
         k1 + k2           k1 + k2

  Hence the ratio of the distances of this point from the positive and
  negative plates respectively is equal to the ratio of the velocities
  of the positive and negative ions.

  The other case we shall consider is the very important one in which
  the velocity of the negative ion is exceedingly large compared with
  the positive; this is the case in flames where, as Gold (_Proc. Roy.
  Soc._ 97, p. 43) has shown, the velocity of the negative ion is many
  thousand times the velocity of the positive; it is also very probably
  the case in all gases when the pressure is low. We may get the
  solution of this case either by putting k1/k2 = 0 in equation (8), or
  independently as follows:--Using the same notation as before, we have

    i = n1k1Xe + n2k2Xe,

     d
    --(n2k2X) = q - [alpha]n1n2,
    dx

    dX
    -- = 4[pi](n1 - n2)e.
    dx

  In this case practically all the current is carried by the negative
  ions so that i = n2k2Xe, and therefore q = [alpha]n1n2.

  Thus

    n2 = i/k2Xe, n1 = qk2Xe/[alpha]i.

  Thus

    dX   4[pi]e²k2qX   4[pi]i
    -- = ----------- - ------,
    dx     [alpha]i     k2X

  or

    dX²  8[pi]e²k2qX²     8[pi]i
    -- - ------------ = - ------.
    dx     [alpha]i         k2

  The solution of this equation is

         [alpha]   i²
    X² = ------- ----- + C[epsilon]^(8[pi]e²k2qx/[alpha]i)
            q    k2²e²

  Here x is measured from the positive electrode; it is more convenient
  in this case, however, to measure it from the negative electrode. If x
  be the distance from the negative electrode at which the electric
  force is X, we have from equation (7)

         [alpha]   i²
    X² = ------- ----- + C¹[epsilon]^(8[pi]e²k2qx/[alpha]i)
            q    k2²e²

  To find the value of C¹ we see by equation (7) that

    d²X²   k1k2    1
    ---  ------- ------ = q - [alpha]n1n2;
    dX²  k1 + k2 8[pi]e

  hence
     _                  _        _
    | dX²   k1k2     1   |^x1   / x1
    | --- ------- ------ |    = |  (q - [alpha]n1n2)dx.
    |_dX  k1 + k2 8[pi]e_|     _/0

  The right hand side of this equation is the excess of ionization over
  recombination in the region extending from the cathode to x1; it must
  therefore, when things are in a steady state, equal the excess of the
  number of negative ions which leave this region over those which enter
  it. The number which leave is i/e and the number which enter is i0/e,
  if it is the current of negative ions coming from unit area of the
  cathode, as hot metal cathodes emit large quantities of negative
  electricity i0 may in some cases be considerable, thus the right hand
  side of equation is (i - i0)/e. When x1 is large dX²/dx = 0; hence we
  have from equation

         [alpha]i(i - i0) k1 + k2
    C¹ = ---------------- -------,
              qk1k2e²        k2

  and since k1 is small compared with k2, we have

         [alpha]i²  /   k2 i - i0                                    \
    X² = --------- (1 + -- ------ [epsilon]^{-8[pi]e²k2·qx/[alpha]·i} ).
          qk2²e²    \   k1    i                                      /

  From the values which have been found for k2 and [alpha], we know that
  8[pi]ek2/[alpha] is a large quantity, hence the second term inside the
  bracket will be very small when eqx is equal to or greater than i;
  thus this term will be very small outside a layer of gas next the
  cathode of such thickness that the number of ions produced on it would
  be sufficient, if they were all utilized for the purpose, to carry the
  current; in the case of flames this layer is exceedingly thin unless
  the current is very large. The value of the electric force in the
  uniform part of the field is equal to i/k2e·[root]([alpha]/q), while
  when i0 = 0, the force at the cathode itself bears to the uniform
  force the ratio of (k1 + k2)^½ to k1^½. As k1 is many thousand times
  k2 the force increases with great rapidity as we approach the cathode;
  this is a very characteristic feature of the passage of electricity
  through flames and hot gases. Thus in an experiment made by H. A.
  Wilson with a flame 18 cm. long, the drop of potential within 1
  centimetre of the cathode was about five times the drop in the other
  17 cm. of the tube. The relation between the current and the potential
  difference when the velocity of the negative ion is much greater than
  the positive is very easily obtained. Since the force is uniform and
  equal to i/k2e·[root]([alpha]/q), until we get close to the cathode
  the fall of potential in this part of the discharge will be very
  approximately equal to i/k2e·[root]([alpha]l/q), where l is the
  distance between the electrodes. Close to the cathode, the electric
  force when i0 is not nearly equal to i is approximately given by the
  equation

            i      /[alpha]\^½
    X = --------- (---------) [epsilon]^{-4[pi]e²k2qx/[alpha]i},
        e(k1k2)^½  \   q   /                                                                            ,

  and the fall of potential at the cathode is equal approximately to

      _[oo]
     /
     |   X dx,
    _/0

  that is to

        i      /[alpha]\^½  [alpha] i
    --------- (---------)  ----------.
    e(k1k2)^½  \   q   /   4[pi]e²k2q

  The potential difference between the plates is the sum of the fall of
  potential in the uniform part of the discharge plus the fall at the
  cathode, hence

         /[alpha]\^½ i   /     i[alpha]²      1      \
    V = (---------) --- ( il + --------- ------------ ).
         \   q   /  ek2  \     4[pi]e²q  [root](k1k2)/

  The fall of potential at the cathode is proportional to the square of
  the current, while the fall in the rest of the circuit is directly
  proportional to the current. In the case of flames or hot gases, the
  fall of potential at the cathode is much greater than that in the rest
  of the circuit, so that in such cases the current through the gas
  varies nearly as the square root of the potential difference. The
  equation we have just obtained is of the form

    V = Ai + Bi²,

  and H. A. Wilson has shown that a relation of this form represents the
  results of his experiments on the conduction of electricity through
  flames.

  The expression for the fall of potential at the cathode is inversely
  proportional to q^(3/2), q being the number of ions produced per cubic
  centimetre per second close to the cathode; thus any increase in the
  ionization at the cathode will diminish the potential fall at the
  cathode, and as practically the whole potential difference between the
  electrodes occurs at the cathode, a diminution in the potential fall
  there will be much more important than a diminution in the electric
  force in the uniform part of the discharge, when the force is
  comparatively insignificant. This consideration explains a very
  striking phenomenon discovered many years ago by Hittorf, who found
  that if he put a wire carrying a bead of a volatile salt into the
  flame, it produced little effect upon the current, unless it were
  placed close to the cathode where it gave rise to an enormous increase
  in the current, sometimes increasing the current more than a
  hundredfold. The introduction of the salt increases very largely the
  number of ions produced, so that q is much greater for a salted flame
  than for a plain one. Thus Hittorf's result coincides with the
  conclusions we have drawn from the theory of this class of conduction.

  The fall of potential at the cathode is proportional to i - i0, where
  i0 is the stream of negative electricity which comes from the cathode
  itself, thus as i0 increases the fall of potential at the cathode
  diminishes and the current sent by a given potential difference
  through the gas increases. Now all metals give out negative particles
  when heated, at a rate which increases very rapidly with the
  temperature, but at the same temperature some metals give out more
  than others. If the cathode is made of a metal which emits large
  quantities of negative particles, (i - i0) will for a given value of i
  be smaller than if the metal only emitted a small number of
  particles; thus the cathode fall will be smaller for the metal with
  the greater emissitivity, and the relation between the potential
  difference and the current will be different in the two cases. These
  considerations are confirmed by experience, for it has been found that
  the current between electrodes immersed in a flame depends to a great
  extent upon the metal of which the electrodes are made. Thus
  Pettinelli (_Acc. dei Lincei_ [5], v. p. 118) found that, _ceteris
  paribus_, the current between two carbon electrodes was about 500
  times that between two iron ones. If one electrode was carbon and the
  other iron, the current when the carbon was cathode and the iron anode
  was more than 100 times the current when the electrodes were reversed.
  The emission of negative particles by some metallic oxides, notably
  those of calcium and barium, has been shown by Wehnelt (_Ann. der
  Phys._ 11, p. 425) to be far greater than that of any known metal, and
  the increase of current produced by coating the cathodes with these
  oxides is exceedingly large; in some cases investigated by Tufts and
  Stark (_Physik. Zeits._, 1908, 5, p. 248) the current was increased
  many thousand times by coating the cathode with lime. No appreciable
  effect is produced by putting lime on the anode.

  _Conduction when all the Ions are of one Sign._--There are many
  important cases in which the ions producing the current come from one
  electrode or from a thin layer of gas close to the electrode, no
  ionization occurring in the body of the gas or at the other electrode.
  Among such cases may be mentioned those where one of the electrodes is
  raised to incandescence while the other is cold, or when the negative
  electrode is exposed to ultra-violet light. In such cases if the
  electrode at which the ionization occurs is the positive electrode,
  all the ions will be positively charged, while if it is the negative
  electrode the ions will all be charged negatively. The theory of this
  case is exceedingly simple. Suppose the electrodes are parallel planes
  at right angles to the axis of x; let X be the electric force at a
  distance x from the electrode where the ionization occurs, n the
  number of ions (all of which are of one sign) at this place per cubic
  centimetre, k the velocity of the ion under unit electric force, e the
  charge on an ion, and i the current per unit area of the electrode.
  Then we have dX/dx = 4[pi]ne, and if u is the velocity of the ion neu
  = i. But u = kX, hence we have kX/4[pi] · dX/dx = i, and since the
  right hand side of this equation does not depend upon x, we get
  kX²/8[pi] = ix + C, where C is a constant to be determined. If l is
  the distance between the plates, and V the potential difference
  between them,
         _                _____  _                      _
        / l        1     /8[pi] |                        |
    V = |   Xdx = ---   / ----- | ( il + C )^3/2 - C^3/2 |.
       _/0         i  \/    k   |_                      _|

  We shall show that when the current is far below the saturation value,
  C is very small compared with il, so that the preceding equation
  becomes

    V² = 8[pi]l³i/k                                  (1).

  To show that for small currents C is small compared with il, consider
  the case when the ionization is confined to a thin layer, thickness d
  close to the electrode, in that layer let n0 be the value of n, then
  we have q = [alpha]n0² + i/ed. If X0 be the value of X when x = 0,
  kX0n0e = i, and,

         kX0²        i²        [alpha]      i²
    C = ----- = ----------- = -------- · --------    (2).
        8[pi]   n0²ke·8[pi]   8[pi]ke²   q + i/ed

  Since [alpha]/8[pi]ke is, as we have seen, less than unity, C will be
  small compared with il, if i/(eq + i/d) is small compared with l. If
  I0 is the saturation current, q = I0/ed, so that the former expression
  = id/(I0 + i), if i is small compared with I0, this expression is
  small compared with d, and therefore _a fortiori_ compared with l, so
  that we are justified in this case in using equation (1).

  From equation (2) we see that the current increases as the square of
  the potential difference. Here an increase in the potential difference
  produces a much greater percentage increase than in conduction through
  metals, where the current is proportional to the potential difference.
  When the ionization is distributed through the gas, we have seen that
  the current is approximately proportional to the square root of the
  potential, and so increases more slowly with the potential difference
  than currents through metals. From equation (1) the current is
  inversely proportional to the cube of the distance between the
  electrodes, so that it falls off with great rapidity as this distance
  is increased. We may note that for a given potential difference the
  expression for the current does not involve q, the rate of production
  of the ions at the electrode, in other words, if we vary the
  ionization the current will not begin to be affected by the strength
  of the ionization until this falls so low that the current is a
  considerable fraction of the saturation current. For the same
  potential difference the current is proportional to k, the velocity
  under unit electric force of the ion which carries the current. As the
  velocity of the negative ion is greater than that of the positive, the
  current when the ionization is confined to the neighbourhood of one of
  the electrodes will be greater when that electrode is made cathode
  than when it is anode. Thus the current will appear to pass more
  easily in one direction than in the opposite.

  Since the ions which carry the current have to travel all the way from
  one electrode to the other, any obstacle which is impervious to these
  ions will, if placed between the electrodes, stop the current to the
  electrode where there is no ionization. A plate of metal will be as
  effectual as one made of a non-conductor, and thus we get the
  remarkable result that by interposing a plate of an excellent
  conductor like copper or silver between the electrode, we can entirely
  stop the current. This experiment can easily be tried by using a hot
  plate as the electrode at which the ionization takes place: then if
  the other electrode is cold the current which passes when the hot
  plate is cathode can be entirely stopped by interposing a cold metal
  plate between the electrodes.

_Methods of counting the Number of Ions._--The detection of the ions and
the estimation of their number in a given volume is much facilitated by
the property they possess of promoting the condensation of water-drops
in dust-free air supersaturated with water vapour. If such air contains
no ions, then it requires about an eightfold supersaturation before any
water-drops are formed; if, however, ions are present C. T. R. Wilson
(_Phil. Trans._ 189, p. 265) has shown that a sixfold supersaturation is
sufficient to cause the water vapour to condense round the ions and to
fall down as raindrops. The absence of the drops when no ions are
present is due to the curvature of the drop combined with the surface
tension causing, as Lord Kelvin showed, the evaporation from a small
drop to be exceeding rapid, so that even if a drop of water were formed
the evaporation would be so great in its early stages that it would
rapidly evaporate and disappear. It has been shown, however (J. J.
Thomson, _Application of Dynamics to Physics and Chemistry_, p. 164;
_Conduction of Electricity through Gases_, 2nd ed. p. 179), that if a
drop of water is charged with electricity the effect of the charge is to
diminish the evaporation; if the drop is below a certain size the effect
the charge has in promoting condensation more than counterbalances the
effect of the surface tension in promoting evaporation. Thus the
electric charge protects the drop in the most critical period of its
growth. The effect is easily shown experimentally by taking a bulb
connected with a piston arranged so as to move with great rapidity. When
the piston moves so as to increase the volume of the air contained in
the bulb the air is cooled by expansion, and if it was saturated with
water vapour before it is supersaturated after the expansion. By
altering the throw of the piston the amount of supersaturation can be
adjusted within very wide limits. Let it be adjusted so that the
expansion produces about a sixfold supersaturation; then if the gas is
not exposed to any ionizing agents very few drops (and these probably
due to the small amount of ionization which we have seen is always
present in gases) are formed. If, however, the bulb is exposed to strong
Röntgen rays expansion produces a dense cloud which gradually falls down
and disappears. If the gas in the bulb at the time of its exposure to
the Röntgen rays is subject to a strong electric field hardly any cloud
is formed when the gas is suddenly expanded. The electric field removes
the charged ions from the gas as soon as they are formed so that the
number of ions present is greatly reduced. This experiment furnishes a
very direct proof that the drops of water which form the cloud are only
formed round the ions.

This method gives us an exceedingly delicate test for the presence of
ions, for there is no difficulty in detecting ten or so raindrops per
cubic centimetre; we are thus able to detect the presence of this number
of ions. This result illustrates the enormous difference between the
delicacy of the methods of detecting ions and those for detecting
uncharged molecules; we have seen that we can easily detect ten ions per
cubic centimetre, but there is no known method, spectroscopic or
chemical, which would enable us to detect a billion (10^12) times this
number of uncharged molecules. The formation of the water-drops round
the charged ions gives us a means of counting the number of ions present
in a cubic centimetre of gas; we cool the gas by sudden expansion until
the supersaturation produced by the cooling is sufficient to cause a
cloud to be formed round the ions, and the problem of finding the number
of ions per cubic centimetre of gas is thus reduced to that of finding
the number of drops per cubic centimetre in the cloud. Unless the drops
are very few and far between we cannot do this by direct counting; we
can, however, arrive at the result in the following way. From the amount
of expansion of the gas we can calculate the lowering produced in its
temperature and hence the total quantity of water precipitated. The
water is precipitated as drops, and if all the drops are the same size
the number per cubic centimetre will be equal to the volume of water
deposited per cubic centimetre, divided by the volume of one of the
drops. Hence we can calculate the number of drops if we know their size,
and this can be determined by measuring the velocity with which they
fall under gravity through the air.

  The theory of the fall of a heavy drop of water through a viscous
  fluid shows that v = (2/9)ga²/[mu], where a is the radius of the drop,
  g the acceleration due to gravity, and [mu] the coefficient of
  viscosity of the gas through which the drop falls. Hence if we know v
  we can deduce the value of a and hence the volume of each drop and the
  number of drops.

  _Charge on Ion._--By this method we can determine the number of ions
  per unit volume of an ionized gas. Knowing this number we can proceed
  to determine the charge on an ion. To do this let us apply an electric
  force so as to send a current of electricity through the gas, taking
  care that the current is only a small fraction of the saturating
  current. Then if u is the sum of the velocities of the positive and
  negative ions produced in the electric field applied to the gas, the
  current through unit area of the gas is neu, where n is the number of
  positive or negative ions per cubic centimetre, and e the charge on an
  ion. We can easily measure the current through the gas and thus
  determine neu; we can determine n by the method just described, and u,
  the velocity of the ions under the given electric field, is known from
  the experiments of Zeleny and others. Thus since the product neu, and
  two of the factors n, u are known, we can determine the other factor
  e, the charge on the ion. This method was used by J. J. Thomson, and
  details of the method will be found in _Phil. Mag._ [5], 46, p. 528;
  [5], 48, p. 547; [6], 5, p. 346. The result of these measurements
  shows that the charge on the ion is the same whether the ionization is
  by Röntgen rays or by the influence of ultra-violet light on a metal
  plate. It is the same whether the gas ionized is hydrogen, air or
  carbonic acid, and thus is presumably independent of the nature of the
  gas. The value of e formed by this method was 3.4 × 10^-10
  electrostatic units.

  H. A. Wilson (_Phil. Mag._ [6], 5, p. 429) used another method. Drops
  of water, as we have seen, condense more easily on negative than on
  positive ions. It is possible, therefore, to adjust the expansion so
  that a cloud is formed on the negative but not on the positive ions.
  Wilson arranged the experiments so that such a cloud was formed
  between two horizontal plates which could be maintained at different
  potentials. The charged drops between the plates were acted upon by a
  uniform vertical force which affected their rate of fall. Let X be the
  vertical electric force, e the charge on the drop, v1 the rate of fall
  of the drop when this force acts, and v the rate of fall due to
  gravity alone. Then since the rate of fall is proportionate to the
  force on the drop, if a is the radius of the drop, and [rho] its
  density, then

    Xe + (4/3)[pi][rho]ga³   v1
    ---------------------- = ---,
        4/3[pi][rho]ga³       v

  or   Xe = (4/3)[pi][rho]ga³(v1 - v)/v.

  But  v = 2/9ga²[rho]/[mu],

  so that

           /            / [mu]³   v^(3/2)(v1 - v)
    Xe = \/ 2.9[pi] -  / ------ · ---------------.
                     \/  g[rho]         v

  Thus if X, v, v1 are known e can be determined. Wilson by this method
  found that e was 3.1 × 10^-10 electrostatic units. A few of the ions
  carried charges 2e or 3e.

  Townsend has used the following method to compare the charge carried
  by a gaseous ion with that carried by an atom of hydrogen in the
  electrolysis of solution. We have

    u/D = Ne/[Pi],

  where D is the coefficient of diffusion of the ions through the gas, u
  the velocity of the ion in the same gas when acted on by unit electric
  force, N the number of molecules in a cubic centimetre of the gas when
  the pressure is [Pi] dynes per square centimetre, and e the charge in
  electrostatic units. This relation is obtained on the hypothesis that
  N ions in a cubic centimetre produce the same pressure as N uncharged
  molecules.

  We know the value of D from Townsend's experiments and the values of u
  from those of Zeleny. We get the following values for Ne × 10^-10:--

    +---------------+---------------------+---------------------+
    |               |      Moist Gas.     |      Moist Gas.     |
    +---------------+----------+----------+----------+----------+
    |      Gas.     | Positive | Negative | Positive | Negative |
    |               |   Ions.  |   Ions.  |   Ions.  |   Ions.  |
    +---------------+----------+----------+----------+----------+
    | Air           |   1.28   |   1.29   |   1.46   |   1.31   |
    | Oxygen        |   1.34   |   1.27   |   1.63   |   1.36   |
    | Carbonic acid |   1.01   |    .87   |    .99   |    .93   |
    | Hydrogen      |   1.24   |   1.18   |   1.63   |   1.25   |
    +---------------+----------+----------+----------+----------+
    |    Mean       |   1.22   |   1.15   |   1.43   |   1.21   |
    +---------------+----------+----------+----------+----------+


  Since 1.22 cubic centimetres of hydrogen at the temperature 15° C. and
  pressure 760 mm. of mercury are liberated by the passage through
  acidulated water of one electromagnetic unit of electricity or 3 ×
  10^10 electrostatic units, and since in one cubic centimetre of the
  gas there are 2.46 N atoms of hydrogen, we have, if E is the charge in
  electrostatic units, on the atom of hydrogen in the electrolysis of
  solutions

    2.46NE = 3 × 10^10,

  or

    NE = 1.22 × 10^10.

  The mean of the values of Ne in the preceding table is 1.24 × 10^10.
  Hence we may conclude that the charge of electricity carried by a
  gaseous ion is equal to the charge carried by the hydrogen atom in the
  electrolysis of solutions. The values of Ne for the different gases
  differ more than we should have expected from the probable accuracy of
  the determination of D and the velocity of the ions: Townsend (_Proc.
  Roy. Soc._ 80, p. 207) has shown that when the ionization is produced
  by Röntgen rays some of the positive ions carry a double charge and
  that this accounts for the values of Ne being greater for the positive
  than for the negative ions. Since we know the value of e, viz. 3.5 ×
  10^-10, and, also Ne, = 1.24 × 10^10, we find N the number of
  molecules in a cubic centimetre of gas at standard temperature and
  pressure to be equal to 3.5 × 10^19. This method of obtaining N is the
  only one which does not involve any assumption as to the shape of the
  molecules and the forces acting between them.

  Another method of determining the charge carried by an ion has been
  employed by Rutherford (_Proc. Roy. Soc._ 81, pp. 141, 162), in which
  the positively electrified particles emitted by radium are made use
  of. The method consists of: (1) Counting the number of [alpha]
  particles emitted by a given quantity of radium in a known time. (2)
  Measuring the electric charge emitted by this quantity in the same
  time. To count the number of the [alpha] particles the radium was so
  arranged that it shot into an ionization chamber a small number of
  [alpha] particles per minute; the interval between the emission of
  individual particles was several seconds. When an [alpha] particle
  passed into the vessel it ionized the gas inside and so greatly
  increased its conductivity; thus, if the gas were kept exposed to an
  electric field, the current through the gas would suddenly increase
  when an [alpha] particle passed into the vessel. Although each [alpha]
  particle produces about thirty thousand ions, this is hardly large
  enough to produce the conductivity appreciable without the use of very
  delicate apparatus; to increase the conductivity Rutherford took
  advantage of the fact that ions, especially negative ones, when
  exposed to a strong electric field, produce other ions by collision
  against the molecules of the gas through which they are moving. By
  suitably choosing the electric field and the pressure in the
  ionization chamber, the 30,000 ions produced by each [alpha] particle
  can be multiplied to such an extent that an appreciable current passes
  through the ionization chamber on the arrival of each [alpha]
  particle. An electrometer placed in series with this vessel will show
  by its deflection when an [alpha] particle enters the chamber, and by
  counting the number of deflections per minute we can determine the
  number of [alpha] particles given out by the radium in that time.
  Another method of counting this number is to let the particles fall on
  a phosphorescent screen, and count the number of scintillations on the
  screen in a certain time. Rutherford has shown that these two methods
  give concordant results.

  The charge of positive electricity given out by the radium was
  measured by catching the [alpha] particles in a Faraday cylinder
  placed in a very highly exhausted vessel, and measuring the charge per
  minute received by this cylinder. In this way Rutherford showed that
  the charge on the [alpha] particle was 9.4 × 10^-10 electrostatic
  units. Now e/m for the [alpha] particle = 5 × 10³, and there is
  evidence that the [alpha] particle is a charged atom of helium; since
  the atomic weight of helium is 4 and e/m for hydrogen is 10^4, it
  follows that the charge on the helium atom is twice that on the
  hydrogen, so that the charge on the hydrogen atom is 4.7 × 10^-10
  electrostatic units.

_Calculation of the Mass of the Ions at Low Pressures._--Although at
ordinary pressures the ion seems to have a very complex structure and to
be the aggregate of many molecules, yet we have evidence that at very
low pressures the structure of the ion, and especially of the negative
one, becomes very much simpler. This evidence is afforded by
determination of the mass of the atom. We can measure the ratio of the
mass of an ion to the charge on the ion by observing the deflections
produced by magnetic and electric forces on a moving ion. If an ion
carrying a charge e is moving with a velocity v, at a point where the
magnetic force is H, a mechanical force acts on the ion, whose direction
is at right angles both to the direction of motion of the ion and to the
magnetic force, and whose magnitude is evH sin [theta], where [theta] is
the angle between v and H. Suppose then that we have an ion moving
through a gas whose pressure is so low that the free path of the ion is
long compared with the distance through which it moves whilst we are
experimenting upon it; in this case the motion of the ion will be free,
and will not be affected by the presence of the gas.

  Since the force is always at right angles to the direction of motion
  of the ion, the speed of the ion will not be altered by the action of
  this force; and if the ion is projected with a velocity v in a
  direction at right angles to the magnetic force, and if the magnetic
  force is constant in magnitude and direction, the ion will describe a
  curve in a plane at right angles to the magnetic force. If [rho] is
  the radius of curvature of this curve, m the mass of the ion,
  mv²/[rho] must equal the normal force acting on the ion, i.e. it must
  be equal to Hev, or [rho] = mv/He. Thus the radius of curvature is
  constant; the path is therefore a circle, and if we can measure the
  radius of this circle we know the value of mv/He. In the case of the
  rapidly moving negative ions projected from the cathode in a highly
  exhausted tube, which are known as _cathode rays_, the path of the
  ions can be readily determined since they make many substances
  luminous when they impinge against them. Thus by putting a screen of
  such a substance in the path of the rays the shape of the path will be
  determined. Let us now suppose that the ion is acted upon by a
  vertical electric force X and is free from magnetic force, if it be
  projected with a horizontal velocity v, the vertical deflection y
  after a time t is ½ × et²/m, or if l is the horizontal distance
  travelled over by the ion in this time we have since l = vt,

          Xe l²
    y = ½ -- --.
          m  v²

  Thus if we measure y and l we can deduce e/mv². From the effect of the
  magnetic force we know e/mv. Combining these results we can find both
  e/m and v.

  [Illustration: FIG. 13.]

  The method by which this determination is carried out in practice is
  illustrated in fig. 13. The cathode rays start from the electrode C in
  a highly exhausted tube, pass through two small holes in the plugs A
  and B, the holes being in the same horizontal line. Thus a pencil of
  rays emerging from B is horizontal and produces a bright spot at the
  far end of the tube. In the course of their journey to the end of the
  tube they pass between the horizontal plates E and D, by connecting
  these plates with an electric battery a vertical electric field is
  produced between E and D and the phosphorescent spot is deflected. By
  measuring this deflection we determine e/mv². The tube is now placed
  in a uniform magnetic field, the lines of magnetic force being
  horizontal and at right angles to the plane of the paper. The magnetic
  force makes the rays describe a circle in the plane of the paper, and
  by measuring the vertical deflection of the phosphorescent patch at
  the end of the tube we can determine the radius of this circle, and
  hence the value of e/mv. From the two observations the value of e/m
  and v can be calculated.

  Another method of finding e/m for the negative ion which is applicable
  in many cases to which the preceding one is not suitable, is as
  follows: Let us suppose that the ion starts from rest and moves in a
  field where the electric and magnetic forces are both uniform, the
  electric force X being parallel to the axis of x, and the magnetic
  force Z parallel to the axis of z; then if x, y, are the co-ordinates
  of the ion at the time t, the equations of motion of the ion are--

      d²x           dy
    m --- = Xe - He -- ,
      dt²           dt

      d²y      dx
    m --- = He --.
      dt²      dt

  The solution of these equations, if x, y, dx/dt, dy/dt all vanish when
  t = 0, is

        Xm   /       / e   \ \
    x = --- {1 - cos( -- Ht ) }
        eH²  \       \ m   / /

        Xm   /e          / e   \ \
    y = --- {-- Ht - sin( -- Ht ) }.
        eH²  \m          \ m   / /

  These equations show that the path of the ion is a cycloid, the
  generating circle of which has a diameter equal to 2Xm/eH², and rolls
  on the line x = 0.

  Suppose now that we have a number of ions starting from the plane x =
  0, and moving towards the plane x = a. The particles starting from x =
  0 describe cycloids, and the greatest distance they can get from the
  plane is equal to the diameter of the generating circle of the
  cycloid, i.e. to 2Xm/eH². (After reaching this distance they begin to
  approach the plane.) Hence if a is less than the diameter of the
  generating circle, all the particles starting from x = 0 will reach
  the plane x = a, if this is unlimited in extent; while if a is greater
  than the diameter of the generating circle none of the particles which
  start from x = 0 will reach the plane x = a. Thus, if x = 0 is a plane
  illuminated by ultra-violet light, and consequently the seat of a
  supply of negative ions, and x = a a plane connected with an
  electrometer, then if a definite electric intensity is established
  between the planes, i.e. if X be fixed, so that the rate of emission
  of negative ions from the illuminated plate is given, and if a is less
  than 2Xm/eH², all the ions which start from x = 0 will reach x = a.
  That is, the rate at which this plane receives an electric charge
  will be the same whether there is a magnetic field between the plate
  or not, but if a is greater than 2Xm/eH², then no particle which
  starts from the plate x = 0 will reach the plate x = a, and this plate
  will receive no charge. Thus the supply of electricity to the plate
  has been entirely stopped by the magnetic field. Thus, on this theory,
  if the distance between the plates is less than a certain value, the
  magnetic force should produce no effect on the rate at which the
  electrometer plate receives a charge, while if the distance is greater
  than this value the magnetic force would completely stop the supply of
  electricity to the plate. The actual phenomena are not so abrupt as
  this theory indicates. We find that when the plates are very near
  together the magnetic force produces a very slight effect, and this an
  increase in the rate of charging of the plate. On increasing the
  distance we come to a stage where the magnetic force produces a great
  diminution in the rate of charging. It does not, however, stop it
  abruptly, there being a considerable range of distance, in which the
  magnetic force diminishes but does not destroy the current. At still
  greater distances the current to the plate under the magnetic force is
  quite inappreciable compared with that when there is no magnetic
  force. We should get this gradual instead of abrupt decay of the
  current if some of the particles, instead of all starting from rest,
  started with a finite velocity; in that case the first particles
  stopped would be those which started from rest. This would be when a =
  2Xm/eH². Thus if we measure the value of a when the magnetic force
  first begins to affect the leak to the electrometer we determine
  2Xm/eH², and as we can easily measure X and H, we can deduce the value
  of m/e.

By these methods Thomson determined the value of e/m for the negative
ions produced when ultra-violet light falls on a metal plate, as well as
for the negative ions produced by an incandescent carbon filament in an
atmosphere of hydrogen (_Phil. Mag._ [5], 48, p. 547) as well as for the
cathode rays. It was found that the value of e/m for the negative ions
was the same in all these cases, and that it was a constant quantity
independent of the nature of the gas from which the ions are produced
and the means used to produce them. It was found, too, that this value
was more than a thousand times the value of e/M, where e is the charge
carried by an atom of hydrogen in the electrolysis of solutions, and M
the mass of an atom of hydrogen. We have seen that this charge is the
same as that carried by the negative ion in gases; thus since e/m is
more than a thousand times e/M, it follows that M must be more than a
thousand times m. Thus the mass of the negative ion is exceedingly small
compared with the mass of the atom of hydrogen, the smallest mass
recognized in chemistry. The production of negative ions thus involves
the splitting up of the atom, as from a collection of atoms something is
detached whose mass is less than that of a single atom. It is important
to notice in connexion with this subject that an entirely different line
of argument, based on the Zeeman effect (see MAGNETO-OPTICS), leads to
the recognition of negatively electrified particles for which e/m is of
the same order as that deduced from the consideration of purely
electrical phenomena. These small negatively electrified particles are
called corpuscles. The latest determinations of e/m for corpuscles
available are the following:--

              Observer.                               e/m.

  Classen (_Ber. deut. phys. Ges._ 6, p. 700)    1.7728 × 10^7
  Bucherer (_Ann. der Phys._, 28, p. 513)        1.763 × 10^7

It follows from electrical theory that when the corpuscles are moving
with a velocity comparable with that of light their masses increase
rapidly with their velocity. This effect has been detected by Kauffmann
(_Gött. Nach._, Nov. 8, 1901), who used the corpuscles shot out from
radium, some of which move with velocities only a few per cent less than
that of light. Other experiments on this point have been made by
Bucherer (_Ann. der Phys._ 28, p. 513).

_Conductivity Produced by Ultra-Violet Light._--So much use has been
made in recent times of ultra-violet light for producing ions that it is
desirable to give some account of the electrical effects produced by
light. The discovery by Hertz (_Wied. Ann._ 31, p. 983) in 1887, that
the incidence of ultra-violet light on a spark gap facilitates the
passage of a spark, led to a series of investigations by Hallwachs,
Hoor, Righi and Stoletow, on the effect of ultra-violet light on
electrified bodies. These researches have shown that a freshly cleaned
metal surface, charged with negative electricity, rapidly loses its
charge, however small, when exposed to ultra-violet light, and that if
the surface is insulated and without charge initially, it acquires a
positive charge under the influence of the light. The magnitude of this
positive charge may be very much increased by directing a blast of air
on the plate. This, as Zeleny (_Phil. Mag._ [5], 45, p. 272) showed, has
the effect of blowing from the neighbourhood of the plate negatively
electrified gas, which has similar properties to the charged gas
obtained by the separation of ions from a gas exposed to Röntgen rays or
uranium radiation. If the metal plate is positively electrified, there
is no loss of electrification caused by ultra-violet light. This has
been questioned, but a very careful examination of the question by
Elster and Geitel (_Wied. Ann._ 57, p. 24) has shown that the apparent
exceptions are due to the accidental exposure to reflected ultra-violet
light of metal surfaces in the neighbourhood of the plate negatively
electrified by induction, so that the apparent loss of charge is due to
negative electricity coming up to the plate, and not to positive
electricity going away from it. The ultra-violet light may be obtained
from an arc-lamp, the effectiveness of which is increased if one of the
terminals is made of zinc or aluminium, the light from these substances
being very rich in ultra-violet rays; it may also be got very
conveniently by sparking with an induction coil between zinc or cadmium
terminals. Sunlight is not rich in ultra-violet light, and does not
produce anything like so great an effect as the arc light. Elster and
Geitel, who have investigated with great success the effects of light on
electrified bodies, have shown that the more electro-positive metals
lose negative charges when exposed to ordinary light, and do not need
the presence of the ultra-violet rays. Thus they found that amalgams of
sodium or potassium enclosed in a glass vessel lose a negative charge
when exposed to daylight, though the glass stops the small amount of
ultra-violet light left in sunlight after its passage through the
atmosphere. If sodium or potassium be employed, or, what is more
convenient, the mercury-like liquid obtained by mixing sodium and
potassium in the proportion of their combining weights, they found that
negative electricity was discharged by an ordinary petroleum lamp. If
the still more electro-positive metal rubidium is used, the discharge
can be produced by the light from a glass rod just heated to redness;
but there is no discharge till the glass is luminous. Elster and Geitel
arrange the metals in the following order for the facility with which
negative electrification is discharged by light: rubidium, potassium,
alloy of sodium and potassium, sodium, lithium, magnesium, thallium,
zinc. With copper, platinum, lead, iron, cadmium, carbon and mercury the
effects with ordinary light are too small to be appreciable. The order
is the same as that in Volta's electro-chemical series. With
ultra-violet light the different metals show much smaller differences in
their power of discharging negative electricity than they do with
ordinary light. Elster and Geitel found that the ratio of the
photo-electric effects of two metals exposed to approximately
monochromatic light depended upon the wave-length of the light,
different metals showing a maximum sensitiveness in different parts of
the spectrum. This is shown by the following table for the alkaline
metals. The numbers in the table are the rates of emission of negative
electricity under similar circumstances. The rate of emission under the
light from a petroleum lamp was taken as unity:--

           Blue.   Yellow.  Orange.   Red.
  Rb       .16      .64      .33      .039
  Na       .37      .36      .14      .009
  K        .57      .07      .04      .002

The table shows that the absorption of light by the metal has great
influence on the photo-electric effect, for while potassium is more
sensitive in blue light than sodium, the strong absorption of yellow
light by sodium makes it more than five times more sensitive to this
light than potassium. Stoletow, at an early period, called attention to
the connexion between strong absorption and photo-electric effects. He
showed that water, which does not absorb to any great extent either the
ultra-violet or visible rays, does not show any photo-electric effect,
while strongly coloured solutions, and especially solutions of
fluorescent substances such as methyl green or methyl violet, do so to a
very considerable extent; indeed, a solution of methyl green is more
sensitive than zinc. Hallwachs (_Wied. Ann._ 37, p. 666) proved that in
liquids showing photo-electric effects there is always strong
absorption; we may, however, have absorption without these effects.
Phosphorescent substances, such as calcium sulphide show this effect, as
also do various specimens of fluor-spar. As phosphorescence and
fluorescence are probably accompanied by a very intense absorption by
the surface layers, the evidence is strong that to get the
photo-electric effects we must have strong absorption of some kind of
light, either visible or ultra-violet.

[Illustration: FIG. 14.]

If a conductor A is placed near a conductor B exposed to ultra-violet
light, and if B is made the negative electrode and a difference of
potential established between A and B, a current of electricity will
flow between the conductors. The relation between the magnitude of the
current and the difference of potential when A and B are parallel plates
has been investigated by Stoletow (_Journal de physique_, 1890, 11, p.
469), von Schweidler (_Wien. Ber._, 1899, 108, p. 273) and Varley
(_Phil. Trans. A._, 1904, 202, p. 439). The results of some of Varley's
experiments are represented in the curves shown in fig. 14, in which the
ordinates are the currents and the abscissae the potentials. It will be
seen that when the pressure is exceedingly low the current is
independent of the potential difference and is equal to the negative
charge carried off in unit time by the corpuscles emitted from the
surface exposed to the light. At higher pressures the current rises far
above these values and increases rapidly with the potential difference.
This is due to the corpuscles emitted by the illuminated surface
acquiring under the electric field such high velocities that when they
strike against the molecules of the gas through which they are passing
they ionize them, producing fresh ions which can carry on additional
current. The relation between the current and the potential difference
in this case is in accordance with the results of the theory of
ionization by collision. The corpuscles emitted from a body under the
action of ultra-violet light start from the surface with a finite
velocity. The velocity is not the same for all the corpuscles, nor
indeed could we expect that it should be: for as Ladenburg has shown
(_Ann. der Phys._, 1903, 12, p. 558) the seat of their emission is not
confined to the surface layer of the illuminated metal but extends to a
layer of finite, though small, thickness. Thus the particles which start
deep down will have to force their way through a layer of metal before
they reach the surface, and in doing so will have their velocities
retarded by an amount depending on the thickness of this layer. The
variation in the velocity of the corpuscles is shown in the following
table, due to Lenard (_Ann. der Phys._, 1902, 8, p. 149).

  +------------------------------------+--------+----------+-----------+
  |                                    | Carbon.| Platinum.| Aluminium.|
  +------------------------------------+--------+----------+-----------+
  | Corpuscles emitted with velocities |        |          |           |
  |    between 12 and 8 × 10^7 cm sec. |  0.000 |   0.000  |   0.004   |
  |    between  8 and 4 × 10^7 cm sec. |  0.049 |   0.155  |   0.151   |
  |    between  4 and 0 × 10^7 cm sec. |  0.67  |   0.65   |   0.49    |
  |                                    |        |          |           |
  | Corpuscles only emitted with the   |        |          |           |
  |   help of an external electric     |  0.28  |   0.21   |   0.35    |
  |   field.                           +--------+----------+-----------|
  |                                    |  1.00  |   1.00   |   1.00    |
  +------------------------------------+--------+----------+-----------+

If the illuminated surface is completely surrounded by an envelope of
the same metal insulated from and completely shielded from the light,
the emission of the negative corpuscles from the illuminated surface
would go on until the potential difference V between this surface and
the envelope became so great that the corpuscles with the greatest
velocity lost their energy before reaching the envelope, i.e. if m is
the mass, e the charge on a corpuscle, v the greatest velocity of
projection, until Ve = ½mv². The values found for V by different
observers are not very consistent. Lenard found that V for aluminium was
about 3 volts and for platinum 2. Millikan and Winchester (_Phil. Mag._,
July 1907) found for aluminium V = .738. The apparatus used by them was
so complex that the interpretation of their results is difficult.

An extremely interesting fact discovered by Lenard is that the velocity
with which the corpuscles are emitted from the metal is independent of
the intensity of the incident light. The quantity of corpuscles
increases with the intensity, but the velocity of the individual
corpuscles does not. It is worthy of notice that in other cases when
negative corpuscles are emitted from metals, as for example when the
metals are exposed to cathode rays, Canal-strahlen, or Röntgen rays, the
velocity of the emitted corpuscles is independent of the intensity of
the primary radiation which excites them. The velocity is not, however,
independent of the nature of the primary rays. Thus when light is used
to produce the emission of corpuscles the velocity, as Ladenburg has
shown, depends on the wave length of the light, increasing as the wave
length diminishes. The velocity of corpuscles emitted under the action
of cathode rays is greater than that of those ejected by light, while
the incidence of Röntgen rays produces the emission of corpuscles moving
much more rapidly than those in the cases already mentioned, and the
harder the primary rays the greater is the velocity of the corpuscles.

The importance of the fact that the velocity and therefore the energy of
the corpuscles emitted from the metal is independent of the intensity of
the incident light can hardly be overestimated. It raises the most
fundamental questions as to the nature of light and the constitution of
the molecules. What is the source of the energy possessed by these
corpuscles? Is it the light, or in the stores of internal energy
possessed by the molecule? Let us follow the consequences of supposing
that the energy comes from the light. Then, since the energy is
independent of the intensity of the light, the electric forces which
liberate the corpuscles must also be independent of that intensity. But
this cannot be the case if, as is usually assumed in the electromagnetic
theory, the wave front consists of a uniform distribution of electric
force without structure, for in this case the magnitude of the electric
force is proportional to the square root of the intensity. On the
emission theory of light a difficulty of this kind would not arise, for
on that theory the energy in a luminiferous particle remains constant as
the particle pursues its flight through space. Thus any process which a
single particle is able to effect by virtue of its energy will be done
just as well a thousand miles away from the source of light as at the
source itself, though of course in a given space there will not be
nearly so many particles to do this process far from the source as there
are close in. Thus, if one of the particles when it struck against a
piece of metal caused the ejection of a corpuscle with a given velocity,
the velocity of emission would not depend on the intensity of the light.
There does not seem any reason for believing that the electromagnetic
theory is inconsistent with the idea that on this theory, as on the
emission theory, the energy in the light wave may instead of being
uniformly distributed through space be concentrated in bundles which
occupy only a small fraction of the volume traversed by the light, and
that as the wave travels out the bundles get farther apart, the energy
in each remaining undiminished. Some such view of the structure of light
seems to be required to account for the fact that when a plate of metal
is struck by a wave of ultra-violet light, it would take years before
the corpuscles emitted from the metal would equal in number the
molecules on the surface of the metal plate, and yet on the ordinary
theory of light each one of these is without interruption exposed to the
action of the light. The fact discovered by E. Ladenburg (_Verh. d.
deutsch. physik. Ges._ 9, p. 504) that the velocity with which the
corpuscles are emitted depends on the wave length of the light suggests
that the energy in each bundle depends upon the wave length and
increases as the wave length diminishes.

These considerations illustrate the evidence afforded by photo-electric
effects on the nature of light; these effects may also have a deep
significance with regard to the structure of matter. The fact that the
energy of the individual corpuscles is independent of the intensity of
the light might be explained by the hypothesis that the energy of the
corpuscles does not come from the light but from the energy stored up in
the molecules of the metal exposed to the light. We may suppose that
under the action of the light some of the molecules are thrown into an
unstable state and explode, ejecting corpuscles; the light in this case
acts only as a trigger to liberate the energy in the atom, and it is
this energy and not that of the light which goes into the corpuscles. In
this way the velocity of the corpuscles would be independent of the
intensity of the light. But it may be asked, is this view consistent
with the result obtained by Ladenburg that the velocity of the
corpuscles depends upon the nature of the light? If light of a definite
wave length expelled corpuscles with a definite and uniform velocity, it
would be very improbable that the emission of the corpuscles is due to
an explosion of the atoms. The experimental facts as far as they are
known at present do not allow us to say that the connexion between the
velocity of the corpuscles and the wave length of the light is of this
definite character, and a connexion such as a gradual increase of
average velocity as the wave length of the light diminishes, would be
quite consistent with the view that the corpuscles are ejected by the
explosion of the atom. For in a complex thing like an atom there may be
more than one system which becomes unstable when exposed to light. Let
us suppose that there are two such systems, A and B, of which B ejects
the corpuscles with the greater velocity. If B is more sensitive to the
short waves, and A to the long ones, then as the wave length of the
light diminishes the proportion of the corpuscles which come from B will
increase, and as these are the faster, the average velocity of the
corpuscles emitted will also increase. And although the potential
acquired by a perfectly insulated piece of metal when exposed to
ultra-violet light would depend only on the velocity of the fastest
corpuscles and not upon their number, in practice perfect insulation is
unattainable, and the potential actually acquired is determined by the
condition that the gain of negative electricity by the metal through
lack of insulation, is equal to the loss by the emission of negatively
electrified corpuscles. The potential acquired will fall below that
corresponding to perfect insulation by an amount depending on the number
of the faster corpuscles emitted, and the potential will rise if the
proportion of the rapidly moving corpuscles is increased, even though
there is no increase in their velocity. It is interesting to compare
other cases in which corpuscles are emitted with the case of
ultra-violet light. When a metal or gas is bombarded by cathode rays it
emits corpuscles and the velocity of these is found to be independent of
the velocity of the cathode rays which excite them; the velocity is
greater than for corpuscles emitted under ultra-violet light. Again,
when bodies are exposed to Röntgen rays they emit corpuscles moving with
a much greater velocity than those excited by cathode rays, but again
the velocity does not depend upon the intensity of the rays although it
does to some extent on their hardness. In the case of cathode and
Röntgen rays, the velocity with which the corpuscles are emitted seems,
as far as we know at present, to vary slightly, but only slightly, with
the nature of the substance on which the rays fall. May not this
indicate that the first effect of the primary rays is to detach a
neutral doublet, consisting of a positive and negative charge, this
doublet being the same from whatever system it is detached? And that the
doublet is unstable and explodes, expelling the negative charge with a
high velocity, and the positive one, having a much larger charge, with a
much smaller velocity, the momentum of the negative charge being equal
to that of the positive.

Up to now we have been considering the effects produced when light is
incident on metals. Lenard found (and the result has been confirmed by
the experiments of J. J. Thomson and Lyman) that certain kinds of
ultra-violet light ionize a gas when they pass through. The type of
ultra-violet light which produces this effect is so easily absorbed that
it is stopped by a layer a few millimetres thick of air at atmospheric
pressure.

_Ionization by Collision._--When the ionization of the gas is produced
by external agents such as Röntgen rays or ultra-violet light, the
electric field produces a current by setting the positive ions moving in
one direction, and the negative ones in the opposite; it makes use of
ions already made and does not itself give rise to ionization. In many
cases, however, such as in electric sparks, there are no external agents
to produce ionization and the electric field has to produce the ions as
well as set them in motion. When the ionization is produced by external
means the smallest electric field is able to produce a current through
the gas; when, however, these external means are absent no current is
produced unless the strength of the electric field exceeds a certain
critical value, which depends not merely upon the nature of the gas but
also upon the pressure and the dimensions of the vessel in which it is
contained. The variation of the electric field required to produce
discharge can be completely explained if we suppose that the ionization
of the gas is produced by the impact with its molecules of corpuscles,
and in certain cases of positive ions, which under the influence of the
electric field have acquired considerable kinetic energy. We have direct
evidence that rapidly moving corpuscles are able to ionize molecules
against which they strike, for the cathode rays consist of such
corpuscles, and these when they pass through a gas produce large amounts
of ionization. Suppose then that we have in a gas exposed to an electric
field a few corpuscles. These will be set in motion by the field and
will acquire an amount of energy in proportion to the product of the
electric force, their charge, and the distance travelled in the
direction of the electric field between two collisions with the
molecules of the gas. If this energy is sufficient to give them the
ionizing property possessed by cathode rays, then when a corpuscle
strikes against a molecule it will detach another corpuscle; this under
the action of the electric field will acquire enough energy to produce
corpuscles on its own account, and so as the corpuscles move through the
gas their number will increase in geometrical progression. Thus, though
there were but few corpuscles to begin with, there may be great
ionization after these have been driven some distance through the gas by
the electric field.

  The number of ions produced by collisions can be calculated by the
  following method. Let the electric force be parallel to the axis of x,
  and let n be the number of corpuscles per unit volume at a place fixed
  by the co-ordinate x; then in unit time these corpuscles will make
  nu/[lambda] collisions with the molecules, if u is the velocity of a
  corpuscle and [lambda] the mean free path of a corpuscle. When the
  corpuscles are moving fast enough to produce ions by collision their
  velocities are very much greater than those they would possess at the
  same temperature if they were not acted on by electrical force, and so
  we may regard the velocities as being parallel to the axis of x and
  determined by the electric force and the mean free path of the
  corpuscles. We have to consider how many of the nu/[lambda] collisions
  which take place per second will produce ions. We should expect that
  the ionization of a molecule would require a certain amount of energy,
  so that if the energy of the corpuscle fell below this amount no
  ionization would take place, while if the energy of the corpuscle were
  exceedingly large, every collision would result in ionization. We
  shall suppose that a certain fraction of the number of collisions
  result in ionization and that this fraction is a function of the
  energy possessed by the corpuscle when it collides against the
  molecules. This energy is proportional to Xe[lambda] when X is the
  electric force, e the charge on the corpuscle, and [lambda] the mean
  free path. If the fraction of collisions which produce ionization is
  [int](Xe[lambda]), then the number of ions produced per cubic
  centimetre per second is [int](Xe[lambda])nu/[lambda]. If the
  collisions follow each other with great rapidity so that a molecule
  has not had time to recover from one collision before it is struck
  again, the effect of collisions might be cumulative, so that a
  succession of collisions might give rise to ionization, though none of
  the collisions would produce an ion by itself. In this case [int]
  would involve the frequency of the collisions as well as the energy of
  the corpuscle; in other words, it might depend on the current through
  the gas as well as upon the intensity of the electric field. We
  shall, however, to begin with, assume that the current is so small
  that this cumulative effect may be neglected.

  Let us now consider the rate of increase, dn/dt, in the number of
  corpuscles per unit volume. In consequence of the collisions,
  [int](Xe[lambda])nu/[lambda] corpuscles are produced per second; in
  consequence of the motion of the corpuscles, the number which leave
  unit volume per second is greater than those which enter it by
  (d/dx)(nu); while in a certain number of collisions a corpuscle will
  stick to the molecule and will thus cease to be a free corpuscle. Let
  the fraction of the number of collisions in which this occurs be
  [beta]. Thus the gain in the number of corpuscles is
  [int](Xe[lambda])nu/[lambda], while the loss is (d/dx)(nu) +
  [beta](nu)/[lambda]; hence

    dn                        nu      d        [beta]nu
    -- = [int](Xe[lambda]) -------- - --(nu) - --------.
    dt                     [lambda]   dx       [lambda]

  When things are in a steady state dn/dt = 0, and we have

    d           1     /                          \
    --(nu) = --------( [int](Xe[lambda]) - [beta] )nu.
    dx       [lambda] \                          /

  If the current is so small that the electrical charges in the gas are
  not able to produce any appreciable variations in the field, X will be
  constant and we get nu = C[epsilon]^{[alpha]x}, where [alpha] =
  {[int](Xe[lambda]) - [beta]}/[lambda]. If we take the origin from
  which we measure x at the cathode, C is the value of nu at the
  cathode, i.e. it is the number of corpuscles emitted per unit area of
  the cathode per unit time; this is equal to i0/e if i0 is the quantity
  of negative electricity coming from unit area of the cathode per
  second, and e the electric charge carried by a corpuscle. Hence we
  have nue = i0[epsilon]^{[alpha]x}. If l is the distance between the
  anode and the cathode, the value of nue, when x = l, is the current
  passing through unit area of the gas, if we neglect the electricity
  carried by negatively electrified carriers other than corpuscles.
  Hence i = i0[epsilon]^{[alpha]l}. Thus the current between the plates
  increases in geometrical progression with the distance between the
  plates.

  By measuring the variation of the current as the distance between the
  plates is increased, Townsend, to whom we owe much of our knowledge on
  this subject, determined the values of [alpha] for different values of
  X and for different pressures for air, hydrogen and carbonic acid gas
  (_Phil. Mag._ [6], 1, p. 198). Since [lambda] varies inversely as the
  pressure, we see that [alpha] may be written in the form p[phi](X/p)
  or [alpha]/X = F(X/p). The following are some of the values of [alpha]
  found by Townsend for air.

    +---------+----------+----------+----------+----------+----------+
    | X Volts | Pressure | Pressure | Pressure | Pressure | Pressure |
    | per cm. | .17 mm.  | .38 mm.  | 1.10 mm. | 2.1 mm.  | 4.1 mm.  |
    +---------+----------+----------+----------+----------+----------+
    |    20   |    .24   |          |          |          |          |
    |    40   |    .65   |    .34   |          |          |          |
    |    80   |   1.35   |    1.3   |     .45  |    .13   |          |
    |   120   |   1.8    |    2.0   |    1.1   |    .42   |    .13   |
    |   160   |   2.1    |    2.8   |    2.0   |    .9    |    .28   |
    |   200   |          |    3.4   |    2.8   |   1.6    |    .5    |
    |   240   |   2.45   |    3.8   |    4.0   |   2.35   |    .99   |
    |   320   |   2.7    |    4.5   |    5.5   |   4.0    |   2.1    |
    |   400   |          |    5.0   |    6.8   |   6.0    |   3.6    |
    |   480   |   3.15   |    5.4   |    8.0   |   7.8    |   5.3    |
    |   560   |          |    5.8   |    9.3   |   9.4    |   7.1    |
    |   640   |   3.25   |    6.2   |   10.6   |  10.8    |   8.9    |
    +---------+----------+----------+----------+----------+----------+

  We see from this table that for a given value of X, [alpha] for small
  pressures increases as the pressure increases; it attains a maximum at
  a particular pressure, and then diminishes as the pressure increases.
  The increase in the pressure increases the number of collisions, but
  diminishes the energy acquired by the corpuscle in the electric field,
  and thus diminishes the change of any one collision resulting in
  ionization. If we suppose the field is so strong that at some
  particular pressure the energy acquired by the corpuscle is well above
  the value required to ionize at each collision, then it is evident
  that increasing the number of collisions will increase the amount of
  ionization, and therefore [alpha], and [alpha] cannot begin to
  diminish until the pressure has increased to such an extent that the
  mean free path of a corpuscle is so small that the energy acquired by
  the corpuscle from the electric field falls below the value when each
  collision results in ionization.

  The value of p, when X is given, for which [alpha] is a maximum, is
  proportional to X; this follows at once from the fact that [alpha] is
  of the form X·F(X/p). The value of X/p for which F(X/p) is a maximum is
  seen from the preceding table to be about 420, when X is expressed in
  volts per centimetre and p in millimetres of mercury. The maximum value
  of F(X/p) is about 1/60. Since the current passing between two planes
  at a distance l apart is i0[epsilon]^{[alpha]l} or
  i0[epsilon]^{XlF(X/p)}, and since the force between the plates is
  supposed to be uniform, Xl is equal to V, the potential between the
  plates; hence the current between the plates is i0[epsilon]^{VlF(X/p)},
  and the greatest value it can have is i0[epsilon]^{V/60}. Thus the
  ratio between the current between the plates when there is ionization
  and when there is none cannot be greater than [epsilon]^{V/60}, when V
  is measured in volts. This result is based on Townsend's experiments
  with very weak currents; we must remember, however, that when the
  collisions are so frequent that the effects of collisions can
  accumulate, [alpha] may have much larger values than when the current
  is small. In some experiments made by J. J. Thomson with intense
  currents from cathodes covered with hot lime, the increase in the
  current when the potential difference was 60 volts, instead of being e
  times the current when there was no ionization, as the preceding theory
  indicates, was several hundred times that value, thus indicating a
  great increase in [alpha] with the strength of the current.

  Townsend has shown that we can deduce from the values of [alpha] the
  mean free path of a corpuscle. For if the ionization is due to the
  collisions with the corpuscles, then unless one collision detaches
  more than one corpuscle the maximum number of corpuscles produced will
  be equal to the number of collisions. When each collision results in
  the production of a corpuscle, [alpha] = 1/[lambda] and is independent
  of the strength of the electric field. Hence we see that the value of
  [alpha], when it is independent of the electric field, is equal to the
  reciprocal of the free path. Thus from the table we infer that at a
  pressure of 17 mm. the mean free path is 1/325 cm.; hence at 1 mm. the
  mean free path of a corpuscle is 1/19 cm. Townsend has shown that this
  value of the mean free path agrees well with the value 1/21 cm.
  deduced from the kinetic theory of gases for a corpuscle moving
  through air. By measuring the values of [alpha] for hydrogen and
  carbonic acid gas Townsend and Kirby (_Phil. Mag._ [6], 1, p. 630)
  showed that the mean free paths for corpuscles in these gases are
  respectively 1/11.5 and 1/29 cm. at a pressure of 1 mm. These results
  again agree well with the values given by the kinetic theory of gases.

  If the number of positive ions per unit volume is m and v is the
  velocity, we have nue+mve = i, where i is the current through unit
  area of the gas. Since nue = i0[epsilon]^nx and i = i0[epsilon]^nl,
  when l is the distance between the plates, we see that

    nu / mv = [epsilon]^(nx) / ([epsilon]^(nl) - [epsilon]^(nx)),

    n    v             [epsilon]^(nx)
    -- = -- · -------------------------------.
    m    u    [epsilon]^(ne) - [epsilon]^(nx)

  Since v/u is a very small quantity we see that n will be less than m
  except when [epsilon]^nl - [epsilon]^nx is small, i.e. except close to
  the anode. Thus there will be an excess of positive electricity from
  the cathode almost up to the anode, while close to the anode there
  will be an excess of negative. This distribution of electricity will
  make the electric force diminish from the cathode to the place where
  there is as much positive as negative electricity, where it will have
  its minimum value, and then increase up to the anode.

  The expression i = i0[epsilon]^[alpha]l applies to the case when there
  is no source of ionization in the gas other than the collisions; if in
  addition to this there is a source of uniform ionization producing q
  ions per cubic centimetre, we can easily show that

                                   qe
    i = i0[epsilon]^{[alpha]l} + -------(e^{[alpha]l} - 1).
                                 [alpha]

  With regard to the minimum energy which must be possessed by a
  corpuscle to enable it to produce ions by collision, Townsend (loc.
  cit.) came to the conclusion that to ionize air the corpuscle must
  possess an amount of energy equal to that acquired by the fall of its
  charge through a potential difference of about 2 volts. This is also
  the value arrived at by H. A. Wilson by entirely different
  considerations. Stark, however, gives 17 volts as the minimum for
  ionization. The energy depends upon the nature of the gas; recent
  experiments by Dawes and Gill and Pedduck (_Phil. Mag._, Aug. 1908)
  have shown that it is smaller for helium than for air, hydrogen, or
  carbonic acid gas.

If there is no external source of ionization and no emission of
corpuscles from the cathode, then it is evident that even if some
corpuscles happened to be present in the gas when the electric field
were applied, we could not get a permanent current by the aid of
collisions made by these corpuscles. For under the electric field, the
corpuscles would be driven from the cathode to the anode, and in a short
time all the corpuscles originally present in the gas and those produced
by them would be driven from the gas against the anode, and if there was
no source from which fresh corpuscles could be introduced into the gas
the current would cease. The current, however, could be maintained
indefinitely if the positive ions in their journey back to the cathode
also produced ions by collisions, for then we should have a kind of
regenerative process by which the supply of corpuscles could be
continually renewed. To maintain the current it is not necessary that
the ionization resulting from the positive ions should be anything like
as great as that from the negative, as the investigation given below
shows a very small amount of ionization by the positive ions will
suffice to maintain the current. The existence of ionization by
collision with positive ions has been proved by Townsend. Another method
by which the current could be and is maintained is by the anode emitting
corpuscles under the impact of the positive ions driven against it by
the electric field. J. J. Thomson has shown by direct experiment that
positively electrified particles when they strike against a metal plate
cause the metal to emit corpuscles (J. J. Thomson, _Proc. Camb. Phil.
Soc._ 13, p. 212; Austin, _Phys. Rev._ 22, p. 312). If we assume that
the number of corpuscles emitted by the plate in one second is
proportional to the energy in the positive ions which strike the plate
in that second, we can readily find an expression for the difference of
potential which will maintain without any external ionization a current
of electricity through the gas. As this investigation brings into
prominence many of the most important features of the electric
discharge, we shall consider it in some detail.

  Let us suppose that the electrodes are parallel plates of metal at
  right angles to the axis of x, and that at the cathode x = 0 and at the
  anode x = d, d being thus the distance between the plates. Let us also
  suppose that the current of electricity flowing between the plates is
  so small that the electrification between the plates due to the
  accumulation of ions is not sufficient to disturb appreciably the
  electric field, which we regard as uniform between the plates, the
  electric force being equal to V/d, where V is the potential difference
  between the plates. The number of positive ions produced per second in
  a layer of gas between the planes x and x+dx is [alpha]nu·dx. Here n is
  the number of corpuscles per unit volume, [alpha] the coefficient of
  ionization (for strong electric field [alpha] = 1/[lambda]', where
  [lambda]' is the mean free path of a corpuscle), and u the velocity of
  a corpuscle parallel to x. We have seen that nu = i0[epsilon]^[alpha]x,
  where i0 is the number of corpuscles emitted per second by unit area of
  the cathode. Thus the number of positive ions produced in the layer is
  [alpha]i0[epsilon]^[alpha]x dx. If these went straight to the cathode
  without a collision, each of them would have received an amount of
  kinetic energy Vex/d when they struck the cathode, and the energy of
  the group of ions would be Vex/d·[alpha]i0[epsilon]^dx dx. The positive
  ions will, however, collide with the molecules of the gas through which
  they are passing, and this will diminish the energy they possess when
  they reach the cathode.

  The diminution in the energy will increase in geometrical proportion
  with the length of path travelled by the ion and will thus be
  proportional to [epsilon]^-[beta]x, [beta] will be proportional to the
  number of collisions and will thus be proportional to the pressure of
  the gas. Thus the kinetic energy possessed by the ions when they reach
  the cathode will be

    [epsilon]^{-[beta]x} · V(ex/d) · [alpha]i0[epsilon]^{[alpha]x} dx,

  and E, the total amount of energy in the positive ions which reach the
  cathode in unit time, will be given by the equation
         _
        /d
    E = | [epsilon]^{-[beta]x} · V(ex/d) · [alpha]i0[epsilon]^{[alpha]x} dx
       _/0
                     _
        Ve[alpha]i0 /d
      = ----------- | [epsilon]^{-([beta]-[alpha])x}·x·dx
             d     _/0

        Ve[alpha]i0  /       1                                           /       1                   d        \ \
      = ----------- { ---------------- - [epsilon]^{-([beta]-[alpha])d} { ----------------- + ---------------- } } (1).
             d       \([beta]-[alpha])²                                  \([beta]-[alpha])²   ([beta]-[alpha])/ /


  If the number of corpuscles emitted by the cathode in unit time is
  proportional to this energy we have i0 = kE, where k is a constant;
  hence by equation (1) we have

        ([beta]-[alpha])²   d
    V = ----------------- · --,
            ke[alpha]       I

  where

    I = 1 - [epsilon]^{-([beta]-[alpha])d} (1 + d([beta] - [alpha])).

  Since both [beta] and [alpha] are proportional to the pressure, I and
  ([beta] - [alpha])²d/[alpha] are both functions of pd, the product of
  the pressure and the spark length, hence we see that V is expressed by
  an equation of the form

        1
    V = -- [int](pd)                                                  (2),
        ke

  where [int](pd) denotes a function of pd, and neither p nor d enter
  into the expression for V except in this product. Thus the potential
  difference required to produce discharge is constant as long as the
  product of the pressure and spark length remains constant; in other
  words, the spark potential is constant as long as the mass of the gas
  between the electrodes is constant. Thus, for example, if we halve the
  pressure the same potential difference will produce a spark of twice
  the length. This law, which was discovered by Paschen for fairly long
  sparks (_Annalen_, 37, p. 79), and has been shown by Carr (_Phil.
  Trans._, 1903) to hold for short ones, is one of the most important
  properties of the electric discharge.

  We see from the expression for V that when ([beta] - [alpha])d is very
  large

    V = ([beta] - [alpha])²d/ke[alpha].

  Thus V becomes infinite when d is infinite. Again when ([beta] -
  [alpha])d is very small we find

    V = 1/ke[alpha]d;

  thus V is again infinite when d is nothing. There must therefore be
  some value of d intermediate between zero and infinity for which V is
  a minimum. This value is got by finding in the usual way the value of
  d, which makes the expression for V given in equation (1) a minimum.
  We find that d must satisfy the equation

                                        /                                            \
    1 = [epsilon]^{-([beta]-[alpha])d} {1 + ([beta] - [alpha])d + ([beta] - [alpha]·d)²}.
                                        \                                            /

  We find by a process of trial and error that ([beta]-[alpha])d = 1.8
  is approximately a solution of this equation; hence the distance for
  minimum potential is 1.8/([beta] - [alpha]). Since [beta] and [alpha]
  are both proportional to the pressure, we see that the critical spark
  length varies inversely as the pressure. If we substitute this value
  in the expression for V we find that [=V], the minimum spark
  potential, is given by

    _   [beta] - [alpha]   2.2
    V = ---------------- · ---.
            [alpha]        ke

  Since [beta] and [alpha] are each proportional to the pressure, the
  minimum potential is independent of the pressure of the gas. On this
  view the minimum potential depends upon the metal of which the cathode
  is made, since k measures the number of corpuscles emitted per unit
  time by the cathode when struck by positive ions carrying unit energy,
  and unless [beta] bears the same ratio to [alpha] for all gases the
  minimum potential will also vary with the gas. The measurements which
  have been made of the "cathode fall of potential," which as we shall
  see is equal to the minimum potential required to produce a spark,
  show that this quantity varies with the material of which the cathode
  is made and also with the nature of the gas. Since a metal plate, when
  bombarded by positive ions, emits corpuscles, the effect we have been
  considering must play a part in the discharge; it is not, however, the
  only effect which has to be considered, for as Townsend has shown,
  positive ions when moving above a certain speed ionize the gas, and
  cause it to emit corpuscles. It is thus necessary to take into account
  the ionization of the positive ions.

  Let m be the number of positive ions per unit volume, and w their
  velocity, the number of collisions which occur in one second in one
  cubic centimetre of the gas will be proportional to mwp, where p is
  the pressure of the gas. Let the number of ions which result from
  these collisions be [gamma]mw; [gamma] will be a function of p and of
  the strength of the electric field. Let as before n be the number of
  corpuscles per cubic centimetre, u their velocity, and [alpha]nu the
  number of ions which result in one second from the collisions between
  the corpuscles and the gas. The number of ions produced per second per
  cubic centimetre is equal to [alpha]nu + [gamma]mw; hence when things
  are in a steady state

    d
    --(nu) = [alpha]nu + [gamma]mw ,
    dx

  and

    e(nu + mw) = i,

  where e is the charge on the ion and i the current through the gas.
  The solution of these equations when the field is uniform between the
  plates, is

    enu =  C[epsilon]^{([alpha]-[gamma])x} - [gamma]i/([alpha] - [gamma]),

    emw = -C[epsilon]^{([alpha]-[gamma])x} + [alpha]i/([alpha] - [gamma]),

  where C is a constant of integration. If there is no emission of
  positive ions from the anode enu = i, when x = d. Determining C from
  this condition we find

                  i          /                                                  \
    enu = ----------------- {[alpha][epsilon]^{([alpha]-[gamma])(x-d)} - [gamma] },
          [alpha] - [gamma]  \                                                  /

             [alpha]i        /                                     \
    emw = ----------------- {1 - [epsilon]^{([alpha]-[gamma])(x-d)} }.
          [alpha] - [gamma]  \                                     /

  If the cathode did not emit any corpuscles owing to the bombardment by
  positive ions, the condition that the charge should be maintained is
  that there should be enough positive ions at the cathode to carry the
  current i.e. that emw = i; when x = 0, the condition gives

            i          /                                               \
    ----------------- {[alpha][epsilon]^{-([alpha]-[gamma])d} - [gamma] } = 0,
    [alpha] - [gamma]  \                                               /

  or

    [epsilon]^{[alpha]d}/[alpha] = [epsilon]^{[gamma]d}/[gamma].

  Since [alpha] and [gamma] are both of the form pf(X/p) and X = V/d, we
  see that V will be a function of pd, in agreement with Paschen's law.
  If we take into account both the ionization of the gas and the
  emission of corpuscles by the metal we can easily show that

                                                                _
    [alpha]-[gamma][epsilon]^{([alpha]-[gamma])d}   k[alpha]Ve |              1
    --------------------------------------------- = ---------- |  ------------------------- -
                  [alpha] - [gamma]                     d      |_ ([beta]+[gamma]-[alpha])²
                                                                                                   _
                                               /           1                         d           \  |
       [epsilon]^{-([beta]+[gamma]-[alpha])d} { ------------------------ + ---------------------- } |,
                                               \([beta]+[gamma]-[alpha])²  [beta]+[gamma]-[alpha]/ _|

  where k and [beta] have the same meaning as in the previous
  investigation. When d is large, [epsilon]^{([alpha]-[gamma])d} is also
  large; hence in order that the left-hand side of this equation should
  not be negative [gamma] must be less than [alpha]/[epsilon]^
  {([alpha]-[gamma])d}; as this diminishes as d increases we see that when
  the sparks are very long discharge will take place, practically as soon
  as [gamma] has a finite value, i.e. as soon as the positive ions begin
  to produce fresh ions by their collisions.

In the preceding investigation we have supposed that the electric field
between the plates was uniform; if it were not uniform we could get
discharges produced by very much smaller differences of potential than
are necessary in a uniform field. For to maintain the discharge it is
not necessary that the positive ions should act as ionizers all along
their path; it is sufficient that they should do so in the neighbourhood
of cathode. Thus if we have a strong field close to the cathode we might
still get the discharge though the rest of the field were comparatively
weak. Such a distribution of electric force requires, however, a great
accumulation of charged ions near the cathode; until these ions
accumulate the field will be uniform. If the uniform field existing in
the gas before the discharge begins were strong enough to make the
corpuscles produce ions by collision, but not strong enough to make the
positive ions act as ionizers, there would be some accumulation of ions,
and the amount of this accumulation would depend upon the number of free
corpuscles originally present in the gas, and upon the strength of the
electric field. If the accumulation were sufficient to make the field
near the cathode so strong that the positive ions could produce fresh
ions either by collision with the cathode or with the gas, the discharge
would pass though the gas; if not, there will be no continuous
discharge. As the amount of the accumulation depends on the number of
corpuscles present in the gas, we can understand how it is that after a
spark has passed, leaving for a time a supply of corpuscles behind it,
it is easier to get a discharge to pass through the gas than it was
before.

[Illustration: Fig. 15.]

The inequality of the electric field in the gas when a continuous
discharge is passing through it is very obvious when the pressure of the
gas is low. In this case the discharge presents a highly differentiated
appearance of which a type is represented in fig. 15. Starting from the
cathode we have a thin velvety luminous glow in contact with the
surface; this glow is often called the "first cathode layer." Next this
we have a comparatively dark space whose thickness increases as the
pressure diminishes; this is called the "Crookes's dark space," or the
"second cathode layer." Next this we have a luminous position called the
"negative glow" or the "third cathode layer." The boundary between the
second and third layers is often very sharply defined. Next to the third
layer we have another dark space called the "Faraday dark space." Next
to this and reaching up to the anode is another region of luminosity,
called the "positive column," sometimes (as in fig. 15, a) continuous,
sometimes (as in fig. 15, b) broken up into light or dark patches called
"striations." The dimensions of the Faraday dark space and the positive
column vary greatly with the current passing through the gas and with
its pressure; sometimes one or other of them is absent. These
differences in appearances are accompanied by great difference in the
strength of the electric field. The magnitude of the electric force at
different parts of the discharge is represented in fig. 16, where the
ordinates represent the electric force at different parts of the tube,
the cathode being on the right. We see that the electric force is very
large indeed between the negative glow and the cathode, much larger than
in any other part of the tube. It is not constant in this region, but
increases as we approach the cathode. The force reaches a minimum either
in the negative glow itself or in the part of the Faraday dark space
just outside, after which it increases towards the positive column. In
the case of a uniform positive column the electric force along it is
constant until we get quite close to the anode, when a sudden change,
called the "anode fall," takes place in the potential.

[Illustration:
  _Discharge in Hydrogen
    Pressure 2.25 m.m. Current 0.568·10^-3 ampere_
FIG. 16.]

The difference of potential between the cathode and the negative glow is
called the "cathode potential fall" and is found to be constant for wide
variations in the pressure of the gas and the current passing through.
It increases, however, considerably when the current through the gas
exceeds a certain critical value, depending among other things on the
size of the cathode. This cathode fall of potential is shown by
experiment to be very approximately equal to the minimum potential
difference. The following table contains a comparison of the
measurements of the cathode fall of potentials in various gases made by
Warburg (_Wied. Ann._, 1887, 31, p. 545, and 1890, 40, p. 1), Capstick
(_Proc. Roy. Society_, 1898, 63, p. 356), and Strutt (_Phil. Trans._,
1900, 193, p. 377), and the measurements by Strutt of the smallest
difference of potential which will maintain a spark through these gases.

  +---------+-----------------------------------------+-----------------+
  |         |          Cathode fall in Volts.         |Least potential  |
  |   Gas.  +-----------------------------+-----------+   difference    |
  |         |     Platinum Electrodes.    |Aluminium  |   required to   |
  |         |                             |Electrodes.|maintain a Spark.|
  +---------+-----------+---------+-------+-----------+-----------------+
  |         |  Warburg. |Capstick.|Strutt.|  Warburg. |     Strutt.     |
  +---------+-----------+---------+-------+-----------+-----------------+
  |Air      |  340-350  |    ..   |   ..  |     ..    |       341       |
  |H2       | about 300 |   298   |   ..  |    168    |     302-308     |
  |O2       |     ..    |   369   |   ..  |     ..    |        ..       |
  |N2       |230 if free|   232   |   ..  |    207    |       251       |
  |         |from oxygen|         |       |           |                 |
  |Hg vapour|    340    |    ..   |   ..  |     ..    |        ..       |
  |Helium   |     ..    |    ..   |  226  |     ..    |     261-326     |
  |H2O      |     ..    |   469   |   ..  |     ..    |        ..       |
  |NH3      |     ..    |   582   |   ..  |     ..    |        ..       |
  +---------+-----------+---------+-------+-----------+-----------------+

Thus in the cases in which the measurements could be made with the
greatest accuracy the agreement between the cathode fall and the minimum
potential difference is very close. The cathode fall depends on the
material of which the terminals are made, as is shown by the following
table due to Mey (_Verh. deutsch. physik. Gesell._, 1903, 5, p. 72).

  +------+---------------------------------------------+
  | Gas. |                  Electrode.                 |
  +------+---+---+---+---+---+---+---+---+---+-----+---+
  |      | Pt| Hg| Ag| Cu| Fe| Zn| Al| Mg| Na| Na-K| K |
  +------+---+---+---+---+---+---+---+---+---+-----+---+
  |O2    |369| ..| ..| ..| ..| ..| ..| ..| ..| ..  | ..|
  |H2    |300| ..|295|280|230|213|190|168|185|169  |172|
  |N2    |232|226| ..| ..| ..| ..| ..|207|178|125  |170|
  |He    |226| ..| ..| ..| ..| ..| ..| ..| 80| 78.5| 69|
  |Argon |167| ..| ..| ..| ..| ..|100| ..| ..| ..  | ..|
  +------+---+---+---+---+---+---+---+---+---+-----+---+

The dependence of the minimum potential required to produce a spark upon
the metal of which the cathode is made has not been clearly established,
some observers being unable to detect any difference between the
potential required to spark between electrodes of aluminium and those of
brass, while others thought they had detected such a difference. It is
only with sparks not much longer than the critical spark length that we
could hope to detect this difference. When the current through the gas
exceeds a certain critical value depending among other things on the
size of the cathode, the cathode fall of potential increases rapidly and
at the same time the thickness of the dark spaces diminishes. We may
regard the part of the discharge between the cathode and the negative
glow as a discharge taking place under minimum potential difference
through a distance equal to the critical spark length. An inspection of
fig. 16 will show that we cannot regard the electric field as constant
even for this small distance; it thus becomes a matter of interest to
know what would be the effect on the minimum potential difference
required to produce a spark if there were sufficient ions present to
produce variations in the electric field analogous to those represented
in fig. 16. If the electric force at a distance x from the cathode were
proportional to [epsilon]^-px we should have a state of things much
resembling the distribution of electric force near the cathode. If we
apply to this distribution the methods used above for the case when the
force was uniform, we shall find that the minimum potential is less and
the critical spark length greater than when the electric force is
uniform.

_Potential Difference required to produce a Spark of given Length._--We
may regard the region between the cathode and the negative glow as a
place for the production of corpuscles, these corpuscles finding their
way from this region through the negative glow. The parts of this glow
towards the anode we may regard as a cathode, from which, as from a hot
lime cathode, corpuscles are emitted. Let us now consider what will
happen to these corpuscles shot out from the negative glow with a
velocity depending on the cathode fall of potential and independent of
the pressure. These corpuscles will collide with the molecules of the
gas, and unless there is an external electric field to maintain their
velocity they will soon come to rest and accumulate in front of the
negative glow. The electric force exerted by this cloud of corpuscles
will diminish the strength of the electric field in the region between
the cathode and the negative glow, and thus tend to stop the discharge.
To keep up the discharge we must have a sufficiently strong electric
field between the negative glow and the anode to remove the corpuscles
from this region as fast as they are sent into it from the cathode. If,
however, there is no production of ions in the region between the
negative glow and the anode, all the ions in this region will have come
from near the cathode and will be negatively charged; this negative
electrification will diminish the electric force on the cathode side of
it and thus tend to stop the discharge. This back electric field could,
however, be prevented by a little ionization in the region between the
anode and glow, for this would afford a supply of positive ions, and
thus afford an opportunity for the gas in this region to have in it as
many positive as negative ions; in this case it would not give rise to
any back electromotive force. The ionization which produces these
positive ions may, if the field is intense, be due to the collisions of
corpuscles, or it may be due to radiation analogous to ultra-violet, or
soft Röntgen rays, which have been shown by experiment to accompany the
discharge. Thus in the most simple conditions for discharge we should
have sufficient ionization to keep up the supply of positive ions, and
an electric field strong enough to keep the velocity of the negative
corpuscle equal to the value it has when it emerges from the negative
glow. Thus the force must be such as to give a constant velocity to the
corpuscle, and since the force required to move an ion with a given
velocity is proportional to the pressure, this force will be
proportional to the pressure of the gas. Let us call this force ap; then
if l is the distance of the anode from the negative glow the potential
difference between these points will be alp. The potential difference
between the negative glow and the cathode is constant and equals c;
hence if V is the potential difference between the anode and cathode,
then V = c + alp, a relation which expresses the connexion between the
potential difference and spark length for spark lengths greater than the
critical distance. It is to be remembered that the result we have
obtained applies only to such a case as that indicated above, where the
electric force is constant along the positive column. Experiments with
the discharge through gases at low pressure show the discharge may take
other forms. Thus the positive column may be striated when the force
along it is no longer uniform, or the positive column may be absent;
the discharge may be changed from one of these forms to another by
altering the current. The relation between the potential and the
distance between the electrodes varies greatly, as we might expect, with
the current passing through the gas.

The connexion between the potential difference and the spark length has
been made the subject of a large number of experiments. The first
measurements were made by Lord Kelvin in 1860 (_Collected Papers on
Electrostatics and Magnetism_, p. 247); subsequent experiments have been
made by Baille (_Ann. de chimie et de physique_, 5, 25, p. 486), Liebig
(_Phil. Mag._ [5], 24, p. 106), Paschen (_Wied. Ann._ 37, p. 79), Peace
(_Proc. Roy. Soc._, 1892, 52, p. 99), Orgler (_Ann. der Phys._ 1, p.
159), Strutt (_Phil. Trans._ 193, p. 377), Bouty (_Comptes rendus_, 131,
pp. 469, 503), Earhart (_Phil. Mag._ [6], 1, p. 147), Carr (_Phil.
Trans._, 1903), Russell (_Phil. Mag._ [5], 64, p. 237), Hobbs (_Phil.
Mag._ [6], 10, p. 617), Kinsley (_Phil. Mag._ [6], 9, 692), Ritter
(_Ann. der Phys._ 14, p. 118). The results of their experiments show
that for sparks considerably longer than the critical spark length, the
relation between the potential difference V and the spark length l may
be expressed when the electrodes are large with great accuracy by the
linear relation V = c + blp, where p is the pressure and c and b are
constants depending on the nature of the gas. When the sparks are long
the term blp is the most important and the sparking potential is
proportional to the spark length. Though there are considerable
discrepancies between the results obtained by different observers, these
indicate that the production of a long spark between large electrodes in
air at atmospheric pressure requires a potential difference of 30,000
volts for each centimetre of spark length. In hydrogen only about half
this potential difference is required, in carbonic acid gas the
potential difference is about the same as in air, while Ritter's
experiments show that in helium only about one-tenth of this potential
difference is required.

In the case when the electric field is not uniform, as for example when
the discharge takes place between spherical electrodes, Russell's
experiments show that the discharge takes place as soon as the maximum
electric force in the field between the electrodes reaches a definite
value, which he found was for air at atmospheric pressure about 38,000
volts per centimetre.

_Very Short Sparks._--Some very interesting experiments on the potential
difference required to produce exceedingly short sparks have been made
by Earhart, Hobbs and Kinsley; the length of these sparks was comparable
with the wave length of sodium light. With sparks of these lengths it
was found that it was possible to get a discharge with less than 330
volts, the minimum potential difference in air. The results of these
observers show that there is no diminution in the minimum potential
difference required to produce discharge until the spark length gets so
small that the average electric force between the electrodes amounts to
about one million volts per centimetre. When the force rises to this
value a discharge takes place even though the potential difference is
much less than 330 volts; in some of Earhart's experiments it was only
about 2 volts. This kind of discharge is determined not by the condition
that the potential difference should have a given value, but that the
electric force should have a given value. Another point in which this
discharge differs from the ordinary one is that it is influenced
entirely by the nature of the electrodes and not by the nature or
pressure of the gas between them, whereas the ordinary discharge is in
many cases not affected appreciably by changes in the metal of the
electrodes, but is always affected by changes in the pressure and
character of the gas between them. Kinsley found that when one of these
small sparks passed between the electrodes a kind of metallic bridge was
formed between them, so that they were in metallic connexion, and that
the distance between them had to be considerably increased before the
bridge was broken. Almy (_Phil. Mag._, Sept. 1908), who used very small
electrodes, was unable to get a discharge with less than the minimum
spark potential even when the spark length was reduced to one-third of
the wave length of sodium light. He suggests that the discharges
obtained with larger electrodes for smaller voltages are due to the
electrodes being dragged together by the electrostatic attraction
between them.

_Constitution of the Electric Spark._--Schuster and Hemsalech (_Phil.
Trans._ 193, p. 189), Hemsalech (_Comptes Rendus_, 130, p. 898; 132, p.
917; _Jour. de Phys._ 3. 9, p. 43, and Schenck, _Astrophy. Jour._ 14, p.
116) have by spectroscopic methods obtained very interesting results
about the constitution of the spark. The method employed by Schuster and
Hemsalech was as follows: Suppose we photograph the spectrum of a
horizontal spark on a film which is on the rim of a wheel rotating about
a horizontal axis with great velocity. If the luminosity travelled with
infinite speed from one electrode to the other, the image on the film
would be a horizontal line. If, however, the speed with which the
luminosity travelled between the electrodes was comparable with the
speed of the film, the line would be inclined to the horizontal, and by
measuring the inclinations we could find the speed at which the
luminosity travelled. In this way Schuster and Hemsalech showed that
when an oscillating discharge passed between metallic terminals in air,
the first spark passes through the air alone, no lines of the metal
appearing in its spectrum. This first spark vaporizes some of the metal
and the subsequent sparks passing mainly through the metallic vapour;
the appearance of the lines in the film shows that the velocity of the
luminous part of the vapour was finite. The velocity of the vapour of
metals of low atomic weight was in general greater than that of the
vapour of heavier metals. Thus the velocity of aluminium vapour was 1890
metres per second, that of zinc and cadmium only about 545. Perhaps the
most interesting point in the investigation was the discovery that the
velocities corresponding to different lines in the spectrum of the same
metal were in some cases different. Thus with bismuth some of the lines
indicated a velocity of 1420 metres per second, others a velocity of
only 550, while one ([lambda] = 3793) showed a still smaller velocity.
These results are in accordance with a view suggested by other phenomena
that many of the lines in a spectrum produced by an electrical discharge
originate from systems formed during the discharge and not from the
normal atom or molecule. Schuster and Hemsalech found that by inserting
a coil with large self induction in the primary circuit they could
obliterate the air lines in the discharge.

Schenck, by observing the appearance presented when an alternating
current, produced by discharging Leyden jars, was examined in a rapidly
rotating mirror, found it showed the following stages: (1) a thin bright
line, followed in some cases at intervals of half the period of the
discharge by fainter lines; (2) bright curved streamers starting from
the negative terminal, and diminishing rapidly in speed as they receded
from the cathode; (3) a diffused glow lasting for a much longer period
than either of the preceding. These constituents gave out quite
different spectra.

The structure of the discharge is much more easily studied when the
pressure of the gas is low, as the various parts which make up the
discharge are more widely separated from each other. We have already
described the general appearance of the discharge through gases at low
pressures (see p. 657). There is, however, one form of discharge which
is so striking and beautiful that it deserves more detailed
consideration. In this type of discharge, known as the striated
discharge, the positive column is made up of alternate bright and dark
patches known as _striations_. Some of these are represented in fig. 17,
which is taken from a paper by De la Rue and Müller (_Phil. Trans._,
1878, Pt. 1). This type of discharge only occurs when the current and
the pressure of the gas are between certain limits. It is most
beautifully shown when a Wehnelt cathode is used and the current is
produced by storage cells, as this allows us to use large currents and
to maintain a steady potential difference between the electrodes. The
striations are in consequence very bright and steady. The facts which
have been established about these striations are as follows: The
distance between the bright parts of the striations is greater at low
pressures than at high; it depends also upon the diameter of the tube,
increasing as the diameter of the tube increases. If the discharge tube
is wide at one place and narrow in another the striations will be
closer together in the narrow parts than in the wide. The distance
between the striations depends on the current through the tube. The
relation is not a very simple one, as an increase of current sometimes
increases while under other circumstances it decreases the distance
between the striations (see Willows, _Proc. Camb. Phil. Soc._ 10, p.
302). The electric force is not uniform along the striated discharge,
but is greater in the bright than in the dark parts of the striation. An
example is shown in fig. 16, due to H. A. Wilson, which shows the
distribution of electric force at every place in a striated discharge.
In experiments made by J. J. Thomson (_Phil. Mag._, Oct. 1909), using a
Wehnelt cathode, the variations in the electric force were more
pronounced than those shown in fig. 16. The electric force in this case
changed so greatly that it actually became negative just on the cathode
side of the bright part of the striation. Just inside the striation on
the anode side it rose to a very high value, then continually diminished
towards the bright side of the next striation when it again increased.
This distribution of electric force implies that there is great excess
of negative electricity at the bright head of the striation, and a small
excess of positive everywhere else. The temperature of the gas is higher
in the bright than in the dark parts of the striations. Wood (_Wied.
Ann._ 49, p. 238), who has made a very careful study of the distribution
of temperature in a discharge tube, finds that in those tubes the
temperature varies in the same way as the electric force, but that this
temperature (which it must be remembered is the average temperature of
all the molecules and not merely of those which are taking part in the
discharge) is by no means high; in no part of the discharge did the
temperature in his experiments exceed 100° C.

[Illustration: FIG. 17.]

_Theory of the Striations._--We may regard the heaping up of the
negative charges at intervals along the discharge as the fundamental
feature in the striations, and this heaping up may be explained as
follows. Imagine a corpuscle projected with considerable velocity from a
place where the electric field is strong, such as the neighbourhood of
the cathode; as it moves towards the anode through the gas it will
collide with the molecules, ionize them and lose energy and velocity.
Thus unless the corpuscle is acted on by a field strong enough to supply
it with the energy it loses by collision, its speed will gradually
diminish. Further, when its energy falls below a certain value it will
unite with a molecule and become part of a negative ion, instead of a
corpuscle; at this stage there will be a sudden and very large
diminution in its velocity. Let us now follow the course of a stream of
corpuscles starting from the cathode and approaching the anode. If the
speed falls off as the stream proceeds, the corpuscles in the rear will
gain on those in front and the density of the stream in the front will
be increased. If at a certain place the velocity receives a sudden check
by the corpuscles becoming loaded with a molecule, the density of the
negative electricity will increase at this place with great rapidity,
and here there will be a great accumulation of negative electricity, as
at the bright head on the cathode side of a striation. Now this
accumulation of negative electricity will produce a large electric force
on the anode side; this will drive corpuscles forward with great
velocity and ionize the gas. These corpuscles will behave like those
shot from the cathode and will accumulate again at some distance from
their origin, forming the bright head of the next striation, when the
process will be repeated. On this view the bright heads of the
striations act like electrodes, and the discharge passes from one bright
head to the next as by a number of stepping stones, and not directly
from cathode to anode. The luminosity at the head of the striations is
due to the recombination of the ions. These ions have acquired
considerable energy from the electric field, and this energy will be
available for supplying the energy radiated away as light. The
recombination of ions which do not possess considerable amounts of
energy does not seem to give rise to luminosity. Thus, in an ionized gas
not exposed to an electric field, although we have recombination between
the ions, we need not have luminosity. We have at present no exact data
as to the amount of energy which must be given to an ion to make it
luminous on recombination; it also certainly varies with the nature of
the ion; thus even with hot Wehnelt cathodes J. J. Thomson has never
been able to make the discharge through air luminous with a potential
less than from 16 to 17 volts. The mercury lamps, however, in which the
discharge passes through mercury vapour are luminous with a potential
difference of about 12 volts. It follows that if the preceding theory be
right the potential difference between two bright striations must be
great enough to make the corpuscles ionize by collision and also to give
enough energy to the ions to make them luminous when they recombine. The
difference of potential between the bright parts of successive
striations has been measured by Hohn (_Phys. Zeit._ 9, p. 558); it
varies with the pressure and with the gas. The smallest value given by
Hohn is about 15 volts. In some experiments made by J. J. Thomson, when
the pressure of the gas was very low, the difference of potential
between two adjacent dark spaces was as low as 3.75 volts.

_The Arc Discharge._--The discharges we have hitherto considered have
been characterized by large potential differences and small currents. In
the arc discharge we get very large currents with comparatively small
potential differences. We may get the arc discharge by taking a battery
of cells large enough to give a potential difference of 60 to 80 volts,
and connecting the cells with two carbon terminals, which are put in
contact, so that a current of electricity flows round the circuit. If
the terminals, while the current is on, are drawn apart, a bright
discharge, which may carry a current of many amperes, passes from one to
the other. This arc discharge, as it is called, is characterized by
intense heat and by the brilliant luminosity of the terminals. This
makes it a powerful source of light. The temperature of the positive
terminal is much higher than that of the negative. According to Violle
(_Comptes Rendus_, 115, p. 1273) the temperature of the tip of the
former is about 3500° C, and that of the latter 2700° C. The temperature
of the arc itself he found to be higher than that of either of its
terminals. As the arc passes, the positive terminal gets hollowed out
into a crater-like shape, but the negative terminal remains pointed.
Both terminals lose weight.

  The appearance of the terminals is shown in fig. 18, given by Mrs
  Ayrton (_Proc. Inst. Elec. Eng._ 28, p. 400); a, b represent the
  terminals when the arc is quiet, and c when it is accompanied by a
  hissing sound. The intrinsic brightness of the positive crater does
  not increase with an increase in the current; an increased current
  produces an increase in the area of the luminous crater, but the
  amount of light given out by each unit of area of luminous surface is
  unaltered. This indicates that the temperature of the crater is
  constant; it is probably that at which carbon volatilizes. W. E.
  Wilson (_Proc. Roy. Soc._ 58, p. 174; 60, p. 377) has shown that at
  pressures of several atmospheres the intrinsic brightness of the
  crater is considerably diminished.

  [Illustration: FIG. 18.]

  [Illustration: FIG. 19.]

  The connexion between V, the potential difference between the
  terminals, and l, the length of the arc, is somewhat analogous to that
  which holds for the spark discharge. Fröhlich (_Electrotech. Zeit._ 4,
  p. 150) gives for this connexion the relation V = m + nl, where m and
  n are constants. Mrs Ayrton (_The Electric Arc_, chap. iv.) finds that
  both m and n depend upon the current passing between the terminals,
  and gives as the relation between V and l, V = [alpha] + [beta]/I +
  ([gamma] + [delta]/I)l, where [alpha], [beta], [gamma], [delta] are
  constants and I the current. The relation between current and
  potential difference was made the subject of a series of experiments
  by Ayrton (_Electrician_, 1, p. 319; xi. p. 418), some of whose
  results are represented in fig. 19. For a quiet arc an increase in
  current is accompanied by a fall in potential difference, while for
  the hissing arc the potential difference is independent of the
  current. The quantities m and n which occur in Fröhlich's equation
  have been determined by several experimenters. For carbon electrodes
  in air at atmospheric pressure m is about 39 volts, varying somewhat
  with the size and purity of the carbons; it is diminished by soaking
  the terminals in salt solution. The value of n given by different
  observers varies considerably, ranging from .76 to 2 volts when l is
  measured in millimetres; it depends upon the current, diminishing as
  the current increases. When metallic terminals are used instead of
  carbons, the value of m depends upon the nature of the metal, m in
  general being larger the higher the temperature at which the metal
  volatilizes. Thus v. Lang (_Wied. Ann._ 31, p. 384) found the
  following values for m in air at atmospheric pressure:--C = 35; Pt =
  27.4; Fe = 25; Ni = 26.18; Cu = 23.86; Ag = 15.23; Zn = 19.86; Cd =
  10.28. Lecher (_Wied. Ann._ 33, p. 609) gives Pt = 28, Fe = 20, Ag =
  8, while Arons (_Wied. Ann._ 31, p. 384) found for Hg the value 12.8;
  in this case the fall of potential along the arc itself was abnormally
  small. In comparing these values it is important to remember that
  Lecher (loc. cit.) has shown that with Fe or Pt terminals the arc
  discharge is intermittent. Arons has shown that this is also the case
  with Hg terminals, but no intermittence has been detected with
  terminals of C, Ag or Cu. The preceding measurements refer to mean
  potentials, and no conclusions as to the actual potential differences
  at any time can be drawn when the discharge is discontinuous, unless
  we know the law of discontinuity. The ease with which an arc is
  sustained depends greatly on the nature of the electrodes; when they
  are brass, zinc, cadmium, or magnesium it is exceedingly difficult to
  get the arc.

  [Illustration: FIG. 20.]

  [Illustration: FIG. 21.]

  The potential difference between the terminals is affected by the
  pressure of the gas. The most extensive series of experiments on this
  point is that made by Duncan, Rowland, and Tod (_Electrician_, 31, p.
  60), whose results are represented in fig. 20. We see from these
  curves that for very short arcs the potential difference increases
  continuously with the pressure, but for longer ones there is a
  critical pressure at which the potential difference is a minimum, and
  that this critical pressure seems to increase with the length of arc.
  The nature of the gas also affects the potential difference. The
  magnitude of this effect may be gathered from the following values
  given by Arons (_Ann. der Phys._ 1, p. 700) for the potential
  difference required to produce an arc 1.5 mm. long, carrying a current
  of 4.5 amperes, between terminals of different metals in air and pure
  nitrogen.

    +-----------+------+-----------+
    | Terminal. | Air. | Nitrogen. |
    +-----------+------+-----------+
    |  Ag       |  21  |     ?     |
    |  Zn       |  23  |     21    |
    |  Cd       |  25  |     21    |
    |  Cu       |  27  |     30    |
    |  Fe       |  29  |     20    |
    |  Pt       |  36  |     30    |
    |  Al       |  39  |     27    |
    |  Pb       |  ..  |     18    |
    |  Mg       |  ..  |     22    |
    +-----------+------+-----------+

  Thus, with the discharge for an arc of given length and current, the
  nature of the terminals is the most important factor in determining
  the potential difference. The effects produced by the pressure and
  nature of the surrounding gas, although quite appreciable, are not of
  so much importance, while in the spark discharge the nature of the
  terminals is of no importance, everything depending upon the nature
  and pressure of the gas.

  The potential gradient in the arc is very far from being uniform. With
  carbon terminals Luggin (_Wien. Ber._ 98, p. 1192) found that, with a
  current of 15 amperes, there was a fall of potential of 33.7 close to
  the anode, and one 8.7 close to the cathode, so that the curve
  representing the distribution of potential between the terminals would
  be somewhat like that shown in fig. 21. We have seen that a somewhat
  analogous distribution of potential holds in the case of conduction
  through flames, though in that case the greatest drop of potential is
  in general at the cathode and not at the anode. The difference between
  the changes of potential at the anode and cathode is not so large with
  Fe and Cu terminals as with carbon ones; with mercury terminals, Arons
  (_Wied. Ann._ 58, p. 73) found the anode fall to be 7.4 volts, the
  cathode fall 5.4 volts.

The case of the arc when the cathode is a pool of mercury and the anode
a metal wire placed in a vessel from which the air has been exhausted is
one which has attracted much attention, and important investigations on
this point have been made by Hewitt (_Electrician_, 52, p. 447), Wills
(_Electrician_, 54, p. 26), Stark, Retschinsky and Schnaposnikoff (_Ann.
der Phys._ 18, p. 213) and Pollak (_Ann. der Phys._ 19, p. 217). In this
arrangement the mercury is vaporized by the heat, and the discharge
which passes through the mercury vapour gives an exceedingly bright
light, which has been largely used for lighting factories, &c. The
arrangement can also be used as a rectifier, for a current will only
pass through it when the mercury pool is the cathode. Thus if such a
lamp is connected with an alternating current circuit, it lets through
the current in one direction and stops that in the other, thus
furnishing a current which is always in one direction.

_Theory of the Arc Discharge._--An incandescent body such as a piece of
carbon even when at a temperature far below that of the terminals in an
arc, emits corpuscles at a rate corresponding to a current of the order
of 1 ampere per square centimetre of incandescent surface, and as the
rate of increase of emission with the temperature is very rapid, it is
probably at the rate of many amperes per square centimetre at the
temperature of the negative carbon in the arc. If then a piece of carbon
were maintained at this temperature by some external means, and used as
a cathode, a current could be sent from it to another electrode whether
the second electrode were cold or hot. If, however, these negatively
electrified corpuscles did not produce other ions either by collision
with the gas through which they move or with the anode, the spaces
between cathode and anode would have a negative charge, which would tend
to stop the corpuscles leaving the cathode and would require a large
potential difference between anode and cathode to produce any
considerable current. If, however, there is ionization either in the gas
or at the anode, the positive ions will diffuse into the region of the
discharge until they are sensibly equal in number to the negative ions.
When this is the case the back electromotive force is destroyed and the
same potential difference will carry a much larger current. The arc
discharge may be regarded as analogous to the discharge between
incandescent terminals, the only difference being that in the arc the
terminals are maintained in the state of incandescence by the current
and not by external means. On this view the cathode is bombarded by
positive ions which heat it to such a temperature that negative
corpuscles sufficient to carry the current are emitted by it. These
corpuscles bombard the anode and keep it incandescent. They ionize also,
either directly by collision or indirectly by heating the anode, the gas
and vapour of the metal of which the anode is made, and produce in this
way the supply of positive ions which keep the cathode hot.

_Discharge from a Point._--A very interesting case of electric discharge
is that between a sharply pointed electrode, such as a needle, and a
metal surface of considerable area. At atmospheric pressures the
luminosity is confined to the immediate neighbourhood of the point. If
the sign of the potential of the point does not change, the discharge is
carried by ions of one sign--that of the charge on the pointed
electrode. The velocity of these ions under a given potential gradient
has been measured by Chattock (_Phil. Mag._ 32, p. 285), and found to
agree with that of the ions produced by Röntgen or uranium radiation,
while Townsend (_Phil. Trans._ 195, p. 259) has shown that the charge on
these ions is the same as that on the ions streaming from the point. If
the pointed electrode be placed at right angles to a metal plane serving
as the other electrode, the discharge takes place when, for a given
distance of the point from the plane, the potential difference between
the electrodes exceeds a definite value depending upon the pressure and
nature of the gas through which the discharge passes; its value also
depends upon whether, beginning with a small potential difference, we
gradually increase it until discharge commences, or, beginning with a
large potential difference, we decrease it until the discharge stops.
The value found by the latter method is less than that by the former.
According to Chattock's measurements the potential difference V for
discharge between the point and the plate is given by the linear
relation V = a + bl, where l is the distance of the point from the plate
and a and b are constants. From v. Obermayer's (_Wien. Ber._ 100, 2, p.
127) experiments, in which the distance l was greater than in
Chattock's, it would seem that the potential for larger distances does
not increase quite so rapidly with l as is indicated by Chattock's
relation. The potential required to produce this discharge is much less
than that required to produce a spark of length l between parallel
plates; thus from Chattock's experiments to produce the point discharge
when l = .5 cm. in air at atmospheric pressure requires a potential
difference of about 3800 volts when the pointed electrode is positive,
while to produce a spark at the same distance between plane electrodes
would require a potential difference of about 15,000 volts. Chattock
showed that with the same pointed electrode the value of the electric
intensity at the point was the same whatever the distance of the point
from the plane. The value of the electric intensity depended upon the
sharpness of the point. When the end of the pointed electrode is a
hemisphere of radius a, Chattock showed that for the same gas at the
same pressure the electric intensity f when discharge takes place is
roughly proportioned to a^-0.8. The value of the electric intensity at
the pointed electrode is much greater than its value at a plane
electrode for long sparks; but we must remember that at a distance from
a pointed electrode equal to a small multiple of the radius of curvature
of its extremity the electric intensity falls very far below that
required to produce discharge in a uniform field, so that the discharge
from a pointed electrode ought to be compared with a spark whose length
is comparable with the radius of curvature of the point. For such short
sparks the electric intensity is very high. The electric intensity
required to produce the discharge from a gas diminishes as the pressure
of the gas diminishes, but not nearly so rapidly as the electric
intensity for long sparks. Here again the discharge from a point is
comparable with short sparks, which, as we have seen, are much less
sensitive to pressure changes than longer ones. The minimum potential at
which the electricity streams from the point does not depend upon the
material of which the point is made; it varies, however, considerably
with the nature of the gas. The following are the results of some
experiments on this point. Those in the first two columns are due to
Röntgen, those in the third and fourth to Precht:--

  +------+-----------------------------+--------------------+
  |      |Discharge Potential. Point +.|   Pressure 760.    |
  | Gas. +--------------+--------------+----------+---------+
  |      | Pressure 205.| Pressure 110.| Point +. | Point -.|
  +------+--------------+--------------+----------+---------+
  |      |     Volts.   |     Volts.   |  Volts.  | Volts.  |
  | H2   |     1296     |     1174     |   2125   |  1550   |
  | O2   |     2402     |     1975     |   2800   |  2350   |
  | CO   |     2634     |     2100     |    ..    |   ..    |
  | CH4  |     2777     |     2317     |    ..    |   ..    |
  | NO   |     3188     |     2543     |    ..    |   ..    |
  | CO2  |     3287     |     2655     |   3475   |  2100   |
  | N2   |      ..      |      ..      |   2600   |  2000   |
  | Air  |      ..      |      ..      |   2750   |  2050   |
  +------+--------------+--------------+----------+---------+

We see from this table that in the case of the discharge from a
positively electrified point the greater the molecular weight of the gas
the greater the potential required for discharge. Röntgen concluded from
his experiments that the discharging potential from a positive point in
different gases at the same pressure varies inversely as the mean free
path of the molecules of the gas. In the same gas, however, at different
pressures the discharging potential does not vary so quickly with the
pressure as does the mean free path. In Precht's experiments, in which
different gases were used, the variations in the discharging potential
are not so great as the variations in the mean free path of the gases.

The current of electrified air flowing from the point when the
electricity is escaping--the well-known "electrical wind"--is
accompanied by a reaction on the point which tends to drive it
backwards. This reaction has been measured by Arrhenius (_Wied. Ann._
63, p. 305), who finds that when positive electricity is escaping from a
point in air the reaction on the point for a given current varies
inversely as the pressure of the gas, and for different gases (air,
hydrogen and carbonic acid) inversely as the square root of the
molecular weight of the gas. The reaction when negative electricity is
escaping is much less. The proportion between the reactions for positive
and negative currents depends on the pressure of the gas. Thus for equal
positive and negative currents in air at a pressure of 70 cm. the
reaction for a positive point was 1.9 times that of a negative one, at
40 cm. pressure 2.6 times, at 20 cm. pressure 3.2 times, at 10.3 cm.
pressure 7 times, and at 5.1 cm. pressure 15 times the reaction for the
negative point. Investigation shows that the reaction should be
proportional to the quotient of the current by the velocity acquired by
an ion under unit potential gradient. Now this velocity is inversely
proportional to the pressure, so that the reaction should on this view
be directly proportional to the pressure. This agrees with Arrhenius'
results when the point is positive. Again, the velocities of an ion in
hydrogen, air and carbonic acid at the same pressure are approximately
inversely proportional to the square roots of their molecular weights,
so that the reaction should be directly proportional to this quantity.
This also agrees with Arrhenius' results for the discharge from a
positive point. The velocity of the negative ion is greater than that of
a positive one under the same potential gradient, so that the reaction
for the negative point should be less than that for a positive one, but
the excess of the positive reaction over the negative is much greater
than that of the velocity of the negative ion over the velocity of the
positive. There is, however, reason to believe that a considerable
condensation takes place around the negative ion as a nucleus after it
is formed, so that the velocity of the negative ion under a given
potential gradient will be greater immediately after the ion is formed
than when it has existed for some time. The measurements which have been
made of the velocities of the ions relate to those which have been some
time in existence, but a large part of the reaction will be due to the
newly-formed ions moving with a greater velocity, and thus giving a
smaller reaction than that calculated from the observed velocity.

With a given potential difference between the point and the neighbouring
conductor the current issuing from the point is greater when the point
is negative than when it is positive, except in oxygen, when it is less.
Warburg (_Sitz. Akad. d. Wissensch. zu Berlin_, 1899, 50, p. 770) has
shown that the addition of a small quantity of oxygen to nitrogen
produces a great diminution in the current from a negative point, but
has very little effect on the discharge from a positive point. Thus the
removal of a trace of oxygen made a leak from a negative point 50 times
what it was before. Experiments with hydrogen and helium showed that
impurities in these gases had a great effect on the current when the
point was negative, and but little when it was positive. This suggests
that the impurities, by condensing round the negative ions as nuclei,
seriously diminish their velocity. If a point is charged up to a high
and rapidly alternating potential, such as can be produced by the
electric oscillations started when a Leyden jar is discharged, then in
hydrogen, nitrogen, ammonia and carbonic acid gas a conductor placed in
the neighbourhood of the point gets a negative charge, while in air and
oxygen it gets a positive one. There are two considerations which are of
importance in connexion with this effect. The first is the velocity of
the ions in the electric field, and the second the ease with which the
ions can give up their charges to the metal point. The greater velocity
of the negative ions would, if the potential were rapidly alternating,
cause an excess of negative ions to be left in the surrounding gas. This
is the case in hydrogen. If, however, the metal had a much greater
tendency to unite with negative than with positive ions, such as we
should expect to be the case in oxygen, this would act in the opposite
direction, and tend to leave an excess of positive ions in the gas.

_The Characteristic Curve for Discharge through Gases._--When a current
of electricity passes through a metallic conductor the relation between
the current and the potential difference is the exceedingly simple one
expressed by Ohm's law; the current is proportional to the potential
difference. When the current passes through a gas there is no such
simple relation. Thus we have already mentioned cases where the current
increased as the potential increased although not in the same
proportion, while as we have seen in certain stages of the arc discharge
the potential difference diminishes as the current increases. Thus the
problem of finding the current which a given battery will produce when
part of the circuit consists of a gas discharge is much more complicated
than when the circuit consists entirely of metallic conductors. If,
however, we measure the potential difference between the electrodes in
the gas when different currents are sent through it, we can plot a
curve, called the "characteristic curve," whose ordinates are the
potential differences between the electrodes in the gas and the
abscissae the corresponding currents. By the aid of this curve we can
calculate the current produced when a given battery is connected up to
the gas by leads of known resistance.

  For let E0 be the electromotive force of the battery, R the resistance
  of the leads, i the current, the potential difference between the
  terms in the gas will be E0 - Ri. Let ABC (fig. 22) be the
  "characteristic curve," the ordinates being the potential difference
  between the terminals in the gas, and the abscissae the current. Draw
  the line LM whose equation is E = E0 - Ri, then the points where this
  line cuts the characteristic curves will give possible values of i and
  E, the current through the discharge tube and the potential difference
  between the terminals. Some of these points may, however, correspond
  to an unstable position and be impossible to realize. The following
  method gives us a criterion by which we can distinguish the stable
  from the unstable positions. If the current is increased by [delta]i,
  the electromotive force which has to be overcome by the battery is
  R[delta]i + dE/di · [delta]i. If R + dE/di is positive there will be
  an unbalanced electromotive force round the circuit tending to stop
  the current. Thus the increase in the current will be stopped and the
  condition will be a stable one. If, however, R + dE/di is negative
  there will be an unbalanced electromotive force tending to increase
  the current still further; thus the current will go on increasing and
  the condition will be unstable. Thus for stability R + dE/di must be
  positive, a condition first given by Kaufmann (_Ann. der Phys._ 11, p.
  158). The geometrical interpretation of this condition is that the
  straight line LM must, at the point where it cuts the characteristic
  curve, be steeper than the tangent to characteristic curve. Thus of
  the points ABC where the line cuts the curve in fig. 22, A and C
  correspond to stable states and B to an unstable one. The state of
  things represented by a point P on the characteristic curve when the
  slope is downward cannot be stable unless there is in the external
  circuit a resistance greater than that represented by the tangent of
  the inclination of the tangent to the curve at P to the horizontal
  axis.

  [Illustration: FIG. 22.]

  If we keep the external electromotive force the same and gradually
  increase the resistance in the leads, the line LM will become steeper
  and steeper. C will move to the left so that the current will
  diminish; when the line gets so steep that it touches the curve at C',
  any further increase in the resistance will produce an abrupt change
  in the current; for now the state of things represented by a point
  near A' is the only stable state. Thus if the BC part of the curve
  corresponded to a luminous discharge and the A part to a dark
  discharge, we see that if the electromotive force is kept constant
  there is a minimum value of the current for the luminous discharge. If
  the current is reduced below this value, the discharge ceases to be
  luminous, and there is an abrupt diminution in the current.

_Cathode Rays._--When the gas in the discharge tube is at a very low
pressure some remarkable phenomena occur in the neighbourhood of the
cathode. These seem to have been first observed by Plücker (_Pogg. Ann._
107, p. 77; 116, p. 45) who noticed on the walls of the glass tube near
the cathode a greenish phosphorescence, which he regarded as due to rays
proceeding from the cathode, striking against the sides of the tube, and
then travelling back to the cathode. He found that the action of a
magnet on these rays was not the same as the action on the part of the
discharge near the positive electrode. Hittorf (_Pogg. Ann._ 136, p. 8)
showed that the agent producing the phosphorescence was intercepted by a
solid, whether conductor or insulator, placed between the cathode and
the sides of the tube. He regarded the phosphorescence as caused by a
motion starting from the cathode and travelling in straight lines
through the gas. Goldstein (_Monat. der Berl. Akad._, 1876, p. 24)
confirmed this discovery of Hittorf's, and further showed that a
distinct, though not very sharp, shadow is cast by a small object placed
near a large plane cathode. This is a proof that the rays producing the
phosphorescence must be emitted almost normally from the cathode, and
not, like the rays of light from a luminous surface, in all directions,
for such rays would not produce a perceptible shadow if a small body
were placed near the plane. Goldstein regarded the phosphorescence as
due to waves in the ether, for whose propagation the gas was not
necessary. Crookes (_Phil. Trans._, 1879, pt. i. p. 135; pt. ii. pp.
587, 661), who made many remarkable researches in this subject, took a
different view. He regarded the rays as streams of negatively
electrified particles projected normally from the cathode with great
velocity, and, when the pressure is sufficiently low, reaching the sides
of the tube, and by their impact producing phosphorescence and heat. The
rays on this view are deflected by a magnet, because a magnet exerts a
force on a charged moving body.

These rays striking against glass make it phosphorescent. The colour of
the phosphorescence depends on the kind of glass; thus the light from soda
glass is a yellowish green, and that from lead glass blue. Many other
bodies phosphoresce when exposed to these rays, and in particular the
phosphorescence of some gems, such as rubies and diamonds, is exceedingly
vivid. The spectrum of the phosphorescent light is generally continuous,
but Crookes showed that the phosphorescence of some of the rare earths,
such as yttrium, gives a spectrum of bright bands, and he founded on this
fact a spectroscopic method of great importance. Goldstein (_Wied. Ann._
54, p. 371) discovered that the haloid salts of the alkali metals change
colour under the rays, sodium chloride, for example, becoming violet. The
coloration is a surface one, and has been traced by E. Wiedemann and
Schmidt (_Wied. Ann._ 54, p. 618) to the formation of a subchloride.
Chlorides of tin, mercury and lead also change colour in the same way. E.
Wiedemann (_Wied. Ann._ 56, p. 201) discovered another remarkable effect,
which he called thermo-luminescence; he found that many bodies after being
exposed to the cathode rays possess for some time the power of becoming
luminous when their temperature is raised to a point far below that at
which they become luminous in the normal state. Substances belonging to
the class called by van 't Hoff solid solutions exhibit this property of
thermo-luminescence to a remarkable extent. They are formed when two
salts, one greatly in excess of the other, are simultaneously precipitated
from a solution. A trace of MnSO4 in CaSO4 shows very brilliant
thermo-luminescence. The impact of cathode rays produces after a time
perceptible changes in the glass. Crookes (_Phil. Trans._ pt. ii. 1879, p.
645) found that after glass has been phosphorescing for some time under
the cathode rays it seems to get tired, and the phosphorescence is not so
bright as it was initially. Thus, for example, when the shadow of a
Maltese cross is thrown on the walls of the tube as in fig. 23, if after
the discharge has been going on for some time the cross is shaken down or
a new cathode used whose line of fire does not cut the cross, the pattern
of the cross will still be seen on the glass, but it will now be brighter
instead of darker than the surrounding portion. The portions shielded by
the cross, not being tired by being made to phosphoresce for a long time,
respond more vigorously to the stimulus than those portions which have not
been protected. Skinner (_Proc. Camb. Phil. Soc._ ix. p. 371) and Thomson
found on the glass which had been exposed to the rays gelatinous
filaments, apparently silica, resulting from the reduction of the glass. A
reducing action was also noticed by Villard (_Journ. de phys._ 3, viii. p.
140) and Wehnelt (_Wied. Ann._ 67, p. 421). It can be well shown by
letting the rays fall on a plate of oxidized copper, when the part struck
by the rays will become bright. The rays heat bodies on which they fall,
and if they are concentrated by using as a cathode a portion of a
spherical surface, the heat at the centre becomes so great that a piece of
platinum wire can be melted or a diamond charred. Measurements of the
heating effects of the rays have been made by Thomson (_Phil. Mag._ [5],
44, p. 293) and Cady (_Ann. der Phys._ 1, p. 678). Crookes (_Phil.
Trans._, 1879, pt. i. p. 152) showed that a vane mounted as in a
radiometer is set in rotation by the rays, the direction of the rotation
being the same as would be produced by a stream of particles proceeding
from the cathode. The movement is not due to the momentum imparted to the
vanes by the rays, but to the difference in temperature between the sides
of the vanes, the rays making the side against which they strike hotter
than the other.

[Illustration: FIG. 23.]

_Effect of a Magnet._--The rays are deflected by a magnet, so that the
distribution of phosphorescence over the glass and the shape and
position of the shadows cast by bodies in the tube are altered by the
proximity of a magnet. The laws of magnetic deflection of these rays
have been investigated by Plücker (_Pogg._ _Ann._ 103, p. 88), Hittorf
(_Pogg. Ann._ 136, p. 213), Crookes (_Phil. Trans._, 1879, pt. 1, p.
557), and Schuster (_Proc. Roy. Soc._ 47, p. 526). The deflection is the
same as that of negatively electrified particles travelling along the
path of the rays. Such particles would in a magnetic field be acted on
by a force at right angles to the direction of motion of the particle
and also to the magnetic force, the magnitude of the force being
proportional to the product of the velocity of the particle, the
magnetic force, and the sine of the angle between these vectors. In this
case we have seen that if the particle is not acted on by an
electrostatic field, the path in a uniform magnetic field is a spiral,
which, if the magnetic force is at right angles to the direction of
projection of the particle, becomes a circle in the plane at right
angles to the magnetic force, the radius being mv/He, where m, v, e are
respectively the mass, velocity and charge on the particle, and H is the
magnetic force. The smaller the difference of potential between the
electrodes of the discharge tube the greater the deflection produced by
a magnetic field of given strength, and as the difference of potential
rapidly increases with diminution of pressure, after a certain pressure
has been passed, the higher the exhaustion of the tube the less the
magnetic deflection of the rays. Birkeland (_Comptes rendus_, 1896, p.
492) has shown that when the discharge is from an induction coil the
cathode rays produced in the tube at any one time are not equally
deflected by a magnet, but that a narrow patch of phosphorescence when
deflected by a magnet is split up into several distinct patches, giving
rise to what Birkeland calls the "magnetic spectrum." Strutt (_Phil.
Mag._ 48, p. 478) has shown that this magnetic spectrum does not occur
if the discharge of a large number of cells is employed instead of the
coil. Thomson (_Proc. Camb. Phil. Soc._ 9, p. 243) has shown that if the
potential difference between the electrodes is kept the same the
magnetic deflection is independent of the nature of the gas filling the
discharge tube; this was tested with gases so different as air,
hydrogen, carbonic acid and methyl iodide.

_Charge of Negative Electricity carried by the Rays._--We have seen that
the rays are deflected by a magnet, as if they were particles charged
with negative electricity. Perrin (_Comptes rendus_, 121, p. 1130)
showed by direct experiment that a stream of negative electricity is
associated with the rays. A modification made by Thomson of Perrin's
experiment is sketched in fig. 24 (_Phil. Mag._ 48, p. 478).

  [Illustration: FIG. 24.]

  The rays start from the cathode A, and pass through a slit in a solid
  brass rod B fitting tightly into the neck of the tube. This rod is
  connected with earth and used as the anode. The rays after passing
  through the slit travel through the vessel C. D and E are two
  insulated metal cylinders insulated from each other, and each having a
  slit cut in its face so as to enable the rays to pass into the inside
  of the inner cylinder, which is connected with an electrometer, the
  outer cylinder being connected with the earth. The two cylinders are
  placed on the far side of the vessel, but out of the direct line of
  fire of the rays. When the rays go straight through the slit there is
  only a very small negative charge communicated to the inner cylinder,
  but when they are deflected by a magnet so that the phosphorescent
  patch falls on the slit in the outer cylinder the inner cylinder
  receives a very large negative charge, the increase coinciding very
  sharply with the appearance of the phosphorescent patch on the slit.
  When the patch is so much deflected by the magnet that it falls below
  the slit, the negative charge in the cylinder again disappears. This
  experiment shows that the cathode rays are accompanied by a stream of
  negative electrification. The same apparatus can be used to show that
  the passage of cathode rays through a gas makes it a conductor of
  electricity. For if the induction coil is kept running and a stream of
  the rays kept steadily going into the inner cylinder, the potential
  of the inner cylinder reaches a definite negative value below which it
  does not fall, however long the rays may be kept going. The cylinder
  reaches a steady state in which the gain of negative electricity from
  the cathode rays is equal to the loss by leakage through the
  conducting gas, the conductivity being produced by the passage of the
  rays through it. If the inner cylinder is charged up initially with a
  greater negative charge than corresponds to the steady state, on
  turning the rays on to the cylinder the negative charge will decrease
  and not increase until it reaches the steady state. The conductivity
  produced by the passage of cathode rays through a gas diminishes
  rapidly with the pressure. When rays pass through a gas at a low
  pressure, they are deflected by an electric field; when the pressure
  of the gas is higher the conductivity it acquires when the cathode
  rays pass through it is so large that the potential gradient cannot
  reach a sufficiently high value to produce an appreciable deflection.

Thus the cathode rays carry a charge of negative electricity; the
experiment described on page 875 (fig. 13) shows that they are deflected
by an electric field as if they were negatively electrified, and are
acted on by a magnetic force in just the way this force would act on a
negatively electrified body moving along the path of the rays. There is
therefore every reason for believing that they are charges of negative
electricity in rapid motion. By measuring the deflection produced by
magnetic and electric fields we can determine the velocity with which
these particles moved and the ratio of the mass of the particle to the
charge carried by it.

We may conclude from the experiments that the value of m/e for the
particles constituting the cathode rays is of the order 1/1.7 × 10^7,
and we have seen that m/e has the same value in all the other cases of
negative ions in a gas at low pressure for which it has been
measured--viz. for the ions produced when ultra-violet light falls on a
metal plate, or when an incandescent carbon filament is surrounded by a
gas at a low pressure, and for the [beta] particles given out by
radio-active bodies. We have also seen that the value of the charge on
the gaseous ion, in all cases in which it has been measured--viz. the
ions produced by Röntgen and uranium radiation, by ultra-violet light,
and by the discharge of electrification from a point--is the same in
magnitude as the charge carried by the hydrogen atom in the electrolysis
of solutions. The mass of the hydrogen alone is, however, 10^-4 times
this charge, while the mass of the carriers of negative electrification
is only 1/1.7 × 10^7 times the charge; hence the mass of the carriers of
the negative electrification is only 1/1700 of the mass of the hydrogen
atom. We are thus, by the study of the electric discharge, forced to
recognize the existence of masses very much smaller than the smallest
mass hitherto recognized.

  Direct determinations of the velocity of the cathode rays have been
  made by J. J. Thomson (_Phil. Mag._ 38, p. 358), who measured the
  interval between the appearance of phosphorescence on two pieces of
  glass placed at a known distance apart, and by Maiorana (_Nuovo
  Cimento_, 4, 6, p. 336) and Battelli and Stefanini (_Phys. Zeit._ 1,
  p. 51), who measured the interval between the arrival of the negative
  charge carried by the rays at two places separated by a known
  distance. The values of the velocity got in this way are much smaller
  than the values got by the indirect methods previously described: thus
  J. J. Thomson at a fairly high pressure found the velocity to be 2 ×
  10^7 cm./sec. Maiorana found values ranging between 10^7 and 6 × 10^7
  cm./sec, and Battelli and Stefanini values ranging from 6 × 10^6 to
  1.2 × 10^7. In these methods it is very difficult to eliminate the
  effect of the interval which elapses between the arrival of the rays
  and the attainment by the means of detection, such as the
  phosphorescence of the glass or the deflection of the electrometer, of
  sufficient intensity to affect the senses.

[Illustration: FIG. 25.]

_Transmission of Cathode Rays through Solids--Lenard Rays._--It was for
a long time believed that all solids were absolutely opaque to these
rays, as Crookes and Goldstein had proved that very thin glass, and even
a film of collodion, cast intensely black shadows. Hertz (_Wied. Ann._
45, p. 28), however, showed that behind a piece of gold-leaf or
aluminium foil an appreciable amount of phosphorescence occurred on the
glass, and that the phosphorescence moved when a magnet was brought
near. A most important advance was next made by Lenard (_Wied. Ann._ 51,
p. 225), who got the cathode rays to pass from the inside of a discharge
tube to the air outside. For this purpose he used a tube like that shown
in fig. 25. The cathode K is an aluminium disc 1.2 cm. in diameter
fastened to a stiff wire, which is surrounded by a glass tube. The anode
A is a brass strip partly surrounding the cathode. The end of the tube
in front of the cathode is closed by a strong metal cap, fastened in
with marine glue, in the middle of which a hole 1.7 mm. in diameter is
bored, and covered with a piece of very thin aluminium foil about .0026
mm. in thickness. The aluminium window is in metallic contact with the
cap, and this and the anode are connected with the earth. The tube is
then exhausted until the cathode rays strike against the window. Diffuse
light spreads from the window into the air outside the tube, and can be
traced in a dark room for a distance of several centimetres. From the
window, too, proceed rays which, like the cathode rays, can produce
phosphorescence, for certain bodies phosphoresce when placed in the
neighbourhood of the window. This effect is conveniently observed by the
platino-cryanide screens used to detect Röntgen radiation. The
properties of the rays outside the tube resemble in all respects those
of cathode rays; they are deflected by a magnet and by an electric
field, they ionize the gas through which they pass and make it a
conductor of electricity, and they affect a photographic plate and
change the colour of the haloid salts of the alkali metals. As, however,
it is convenient to distinguish between cathode rays outside and inside
the tube, we shall call the former Lenard rays. In air at atmospheric
pressure the Lenard rays spread out very diffusely. If the aluminium
window, instead of opening into the air, opens into another tube which
can be exhausted, it is found that the lower the pressure of the gas in
this tube the farther the rays travel and the less diffuse they are. By
filling the tube with different gases Lenard showed that the greater the
density of the gas the greater is the absorption of these rays. Thus
they travel farther in hydrogen than in any other gas at the same
pressure. Lenard showed, too, that if he adjusted the pressure so that
the density of the gas in this tube was the same--if, for example, the
pressure when the tube was filled with oxygen was 1/16 of the pressure
when it was filled with hydrogen--the absorption was constant whatever
the nature of the gas. Becker (_Ann. der Phys._ 17, p. 381) has shown
that this law is only approximately true, the absorption by hydrogen
being abnormally large, and by the inert monatomic gases, such as helium
and argon, abnormally small. The distance to which the Lenard rays
penetrate into this tube depends upon the pressure in the discharge
tube; if the exhaustion in the latter is very high, so that there is a
large potential difference between the cathode and the anode, and
therefore a high velocity for the cathode rays, the Lenard rays will
penetrate farther than when the pressure in the discharge tube is higher
and the velocity of the cathode rays smaller. Lenard showed that the
greater the penetrating power of his rays the smaller was their magnetic
deflection, and therefore the greater their velocity; thus the greater
the velocity of the cathode rays the greater is the velocity of the
Lenard rays to which they give rise. For very slow cathode rays the
absorption by different gases departs altogether from the density law,
so much so that the absorption of these rays by hydrogen is greater than
that by air (Lenard, _Ann. der Phys._ 12, p. 732). Lenard (_Wied. Ann._
56, p. 255) studied the passage of his rays through solids as well as
through gases, and arrived at the very interesting result that the
absorption of a substance depends only upon its density, and not upon
its chemical composition or physical state; in other words, the amount
of absorption of the rays when they traverse a given distance depends
only on the quantity of matter they cut through in the distance.
McClelland (_Proc. Roy. Soc._ 61, p. 227) showed that the rays carry a
charge of negative electricity, and M'Lennan measured the amount of
ionization rays of given intensity produced in different gases, finding
that if the pressure is adjusted so that the density of the different
gases is the same the number of ions per cubic centimetre is also the
same. In this case, as Lenard has shown, the absorption is the same, so
that with the Lenard rays, as with uranium and probably with Röntgen
rays, equal absorption corresponds to equal ionization. A convenient
method for producing Lenard rays of great intensity has been described
by Des Coudres (_Wied. Ann._ 62, p. 134).

_Diffuse Reflection of Cathode Rays._--When cathode rays fall upon a
surface, whether of an insulator or a conductor, cathode rays start from
the surface in all directions. This phenomenon, which was discovered by
Goldstein (_Wied. Ann._ 62, p. 134), has been investigated by Starke
(_Wied. Ann._ 66, p. 49; _Ann. der Phys._ 111, p. 75), Austin and Starke
(_Ann. der Phys._ 9, p. 271), Campbell-Swinton (_Proc. Roy. Soc._ 64, p.
377), Merritt (_Phys. Rev._ 7, p. 217) and Gehrcke (_Ann. der Phys._ 8,
p. 81); it is often regarded as analogous to the diffuse reflection of
light from such a surface as gypsum, and is spoken of as the diffuse
reflection of the cathode rays. According to Merritt and Austin and
Starke the deviation in a magnetic field of these reflected rays is the
same as that of the incident rays. The experiments, however, were
confined to rays reflected so that the angle of reflection was nearly
equal to that of incidence. Gehrcke showed that among the reflected rays
there were a large number which had a much smaller velocity than the
incident ones. According to Campbell-Swinton the "diffuse" reflection is
accompanied by a certain amount of "specular" reflection. Lenard, who
used slower cathode rays than Austin and Starke, could not detect in the
scattered rays any with velocities comparable with that of the incident
rays; he obtained copious supplies of slow rays whose speed did not
depend on the angle of incidence of the primary rays (_Ann. der Phys._
15, p. 485). When the angle of incidence is very oblique the surface
struck by the rays gets positively charged, showing that the secondary
rays are more numerous than the primary.

_Repulsion of two Cathode Streams._--Goldstein discovered that if in a
tube there are two cathodes connected together, the cathodic rays from
one cathode are deflected when they pass near the other. Experiments
bearing on this subject have been made by Crookes and Wiedemann and
Ebert. The phenomena may be described by saying that the repulsion of
the rays from a cathode A by a cathode B is only appreciable when the
rays from A pass through the Crookes dark space round B. This is what we
should expect if we remember that the electric field in the dark space
is far stronger than in the rest of the discharge, and that the gas in
the other parts of the tube is rendered a conductor by the passage
through it of the cathode rays, and therefore incapable of transmitting
electrostatic repulsion.

Scattering of the Negative Electrodes.--In addition to the cathode rays,
portions of metal start normally from the cathode and form a metallic
deposit on the walls of the tube. The amount of this deposit varies very
much with the metal. Crookes (_Proc. Roy. Soc._ 50, p. 88) found that
the quantities of metal torn from electrodes of the same size, in equal
times, by the same current, are in the order Pd, Au, Ag, Pb, Sn, Pt, Cu,
Cd, Ni, In, Fe.... In air there is very little deposit from an Al
cathode, but it is abundant in tubes filled with the monatomic gases,
mercury vapour, argon or helium. The scattering increases as the density
of the gas diminishes. The particles of metal are at low pressures
deflected by a magnet, though not nearly to the same extent as the
cathode rays. According to Grandquist, the loss of weight of the cathode
in a given time is proportional to the square of the current; it is
therefore not, like the loss of the cathode in ordinary electrolysis,
proportional to the quantity of current which passes through it.

[Illustration: FIG. 26.]

_Positive Rays or "Canalstrahlen."_--Goldstein (_Berl. Sitzungsb._ 39,
p. 691) found that with a perforated cathode certain rays occurred
behind the cathode which were not appreciably deflected by a magnet;
these he called Canalstrahlen, but we shall, for reasons which will
appear later, call them "positive rays."

Their appearance is well shown in fig. 26, taken from a paper by Wehnelt
(_Wied. Ann._ 67, p. 421) in which they are represented at B. Goldstein
found that their colour depends on the gas in which they are formed,
being gold-colour in air and nitrogen, rose-colour in hydrogen,
yellowish rose in oxygen, and greenish gray in carbonic acid.

The colour of the luminosity due to positive rays is not in general the
same as that due to anode rays; the difference is exceptionally well
marked in helium, where the cathode ray luminosity is blue while that
due to the positive rays is red. The luminosity produced when the rays
strike against solids is also quite distinct. The cathode rays make the
body emit a continuous spectrum, while the spectrum produced by the
positive rays often shows bright lines. Thus lithium chloride under
cathode rays gives out a steely blue light and the spectrum is
continuous, while under the positive rays the salt gives out a brilliant
red light and the spectrum shows the red helium line. It is remarkable
that the lines on the spectra of the alkali metals are much more easily
produced when the positive rays fall on the oxide of the metal than when
they fall on the metal itself. Thus when the positive rays fall on a
pool of the liquid alloy of sodium and potassium the specks of oxide on
the surface shine with a bright yellow light while the untarnished part
of the surface is quite dark.

W. Wien (_Wied. Ann._ 65, p. 445) measured the values of e/m for the
particles forming the positive rays. Other measurements have been made
by Ewers (_Wied. Ann._ 69, p. 167) and J. J. Thomson (_Phil. Mag._ 13,
p. 561). The differences between the values of e/m for the cathode and
positive rays are very remarkable. For cathode rays whose velocity does
not approach that of light, e/m is always equal to 1.7 × 10^8, while for
the positive rays the greatest value of this quantity yet observed is
10^4, which is also the value of e/m for the hydrogen ions in the
electrolysis of dilute solutions. In some experiments made by J. J.
Thomson (_Phil. Mag._, 14, p. 359) it was found that when the pressure
of the gas was not too low the bright spot produced by the impact of a
pencil of these rays on a phosphorescent screen is deflected by electric
and magnetic forces into a continuous band extending on both sides of
the undeflected position. The portion on one side is in general much
fainter than that on the other. The direction of this deflection shows
that it is produced by particles charged with negative electricity,
while the brighter band is due to particles charged with positive
electricity. The negatively electrified particles which produce the band
c.c are not corpuscles, for from the electric and magnetic deflections
we can find the value of e/m. As this proves to be equal to 10^4, we see
that the mass of the carrier of the negative charge is comparable with
that of an atom, and so very much greater than that of a corpuscle. At
very low pressures part of the phosphorescence disappears, while the
upper portion breaks up into two patches (fig. 27). For one of these the
maximum value of e/m is 10^4 and for the other 5 × 10³. At low pressures
the appearance of the patches and the values of e/m are the same whether
the tube is filled originally with air, hydrogen or helium. In some of
the experiments the tube was exhausted until the pressure was too low to
allow the discharge to pass. A very small quantity of the gas under
investigation was then admitted into the tube, just sufficient to allow
the discharge to pass, and the deflection of the phosphorescent patch
measured. The following gases were admitted into the tube, air, carbonic
oxide, oxygen, hydrogen, helium, argon and neon, but whatever the gas
the appearance of the phosphorescence was the same; in every case there
were two patches, for one of which e/m = 10^4 and for the other e/m =
5 × 10³. In helium at higher pressures another patch was observed, for
which e/m = 2.5 × 10^8. The continuous band into which the
phosphorescent spot is drawn out when the pressure is not exceedingly
low, which involves the existence of particles for which the mean value
of e/m varies from zero to 10^4, can be explained as follows. The rays
on their way to the phosphorescent screen have to pass through gas which
is ionized by the passage through it of the positive rays; this gas will
therefore contain free corpuscles. The particles which constitute the
rays start with a charge of positive electricity. Some of these
particles in their journey through the gas attract a corpuscle whose
negative charge neutralizes the positive charge on the particle. The
particles when in this neutral state may be ionized by collision and
reacquire a positive charge, or by attracting another particle may
become negatively charged, and this process may be repeated several
times on their journey to the phosphorescent screen. Thus some of the
particles, instead of being positively charged for the whole of the time
they are exposed to the electric and magnetic forces, may be for a part
of that time without a charge or even have a negative charge. The
deflection of a particle is proportional to the average value of its
charge whilst under the influence of the deflecting forces. Thus if a
particle is without a charge for a part of the time, its deflection will
be less than that of a particle which has retained its positive charge
for the whole of its journey, while the few particles which have a
negative charge for a longer time than they have a positive will be
deflected in the opposite direction to the main portion and will produce
the tail (fig. 27).

[Illustration: Fig. 27.]

A similar explanation will apply to the positive rays discovered by
Villard (_Comptes rendus_, 143, p. 674) and J. J. Thomson (_Phil. Mag._
13, p. 359), which travel in the opposite direction to the rays we have
been considering, i.e. they travel away from the cathode and in the
direction of the cathode's rays; these rays are sometimes called
"retrograde" rays. These as far as has been observed have always the
same maximum value of e/m, i.e. 10^4, and there are a considerable
number of negative ones always mixed with them. The maximum velocity of
both the positive and retrograde rays is about 2 × 10^8 cm./sec. and
varies very little with the potential difference between the electrodes
in the tube in which they are produced (J. J. Thomson, _Phil. Mag._,
Dec. 1909).

The positive rays show, when the pressure is not very low, the line
spectrum of the gas through which they pass. An exceedingly valuable set
of observations on this point have been made by Stark and his pupils
(_Physik. Zeit._ 6, p. 892; _Ann. der Phys._ 21, pp. 40, 457). Stark has
shown that in many gases, notably hydrogen, the spectrum shows the
Doppler effect, and he has been able to calculate in this way the
velocity of the positive rays.

_Anode Rays._--Gehrcke and Reichenhein (_Ann. der Phys._ 25, p. 861)
have found that when the anode consists of a mixture of sodium and
lithium chloride raised to a high temperature either by the discharge
itself or by an independent heating circuit, very conspicuous rays come
from the anode when the pressure of the gas in the discharge tube is
very low, and a large coil is used to produce the discharge. The
determination of e/m for these rays showed that they are positively
charged atoms of sodium or lithium, moving with very considerable
velocity; in some of Gehrcke's experiments the maximum velocity was as
great as 1.8 × 10^7 cm./sec. though the average was about 10^7 cm./sec.
These velocities are less than those of the positive rays whose maximum
velocity is about 2 × 10^8 cm./sec.     (J. J. T.)


FOOTNOTES:

  [1] The values for nickel and bismuth given in the table are much
    higher than later values obtained with pure electrolytic nickel and
    bismuth.

  [2] The value here given, namely 12.885, for the electric
    mass-resistivity of liquid mercury as determined by Matthiessen is
    now known to be too high by nearly 1%. The value at present accepted
    is 12.789 ohms per metre-gramme at 0° C.

  [3] The value (1630) here given for hard-drawn copper is about ¼%
    higher than the value now adopted, namely, 1626. The difference is
    due to the fact that either Jenkin or Matthiessen did not employ
    precisely the value at present employed for the density of hard-drawn
    and annealed copper in calculating the volume-resistivities from the
    mass-resistivities.

  [4] Matthiessen's value for nickel is much greater than that obtained
    in more recent researches. (See Matthiessen and Vogt, _Phil. Trans._,
    1863, and J. A. Fleming, _Proc. Roy. Soc._, December 1899.)

  [5] Matthiessen's value for mercury is nearly 1% greater than the
    value adopted at present as the mean of the best results, namely
    94,070.

  [6] The samples of silver, copper and nickel employed for these tests
    were prepared electrolytically by Sir J. W. Swan, and were
    exceedingly pure and soft. The value for volume-resistivity of nickel
    as given in the above table (from experiments by J. A. Fleming,
    _Proc. Roy. Soc._, December 1899) is much less (nearly 40%) than the
    value given by Matthiessen's researches.

  [7] The electrolytic bismuth here used was prepared by Hartmann and
    Braun, and the resistivity taken by J. A. Fleming. The value is
    nearly 20% less than that given by Matthiessen.

  [8] In 1899 a committee was formed of representatives from eight of
    the leading manufacturers of insulated copper cables with delegates
    from the Post Office and Institution of Electrical Engineers, to
    consider the question of the values to be assigned to the resistivity
    of hard-drawn and annealed copper. The sittings of the committee were
    held in London, the secretary being A. H. Howard. The values given in
    the above paragraphs are in accordance with the decision of this
    committee, and its recommendations have been accepted by the General
    Post Office and the leading manufacturers of insulated copper wire
    and cables.

  [9] Platinoid is an alloy introduced by Martino, said to be similar
    in composition to German silver, but with a little tungsten added. It
    varies a good deal in composition according to manufacture, and the
    resistivity of different specimens is not identical. Its electric
    properties were first made known by J. T. Bottomley, in a paper read
    at the Royal Society, May 5, 1885.

  [10] An equivalent gramme molecule is a weight in grammes equal
    numerically to the chemical equivalent of the salt. For instance, one
    equivalent gramme molecule of sodium chloride is a mass of 58.5
    grammes. NaCl = 58.5.

  [11] F. Kohlrausch and L. Holborn, _Das Leitvermögen der Elektrolyte_
    (Leipzig, 1898).

  [12] It should be noticed that the velocities calculated in
    Kohlrausch's theory and observed experimentally are the average
    velocities, and involve both the factors mentioned above; they
    include the time wasted by the ions in combination with each other,
    and, except at great dilution, are less than the velocity with which
    the ions move when free from each other.





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