THE ELECTRIC AND LUMINIFEROUS iREUlUM. 
815 
force H at right angles to itself, there must' be an electric force at riglit angles to 
both, acting on each particular ion, of which the intensity is aH.* For example if 
H were 10® c.g.s., this electric force would be 10®a c.g.s. or 10“®a volts; in the rough 
estimates of the last paragraph it would be of order 10 “b’, as compared with a slope 
of potential along the conductor of 164t; therefore it is quite amenable to observation, 
so that we must consider it more closely. As there are also an equal number of 
negative ions moving in the opposite direction, they must give rise to an opposite 
electric force acting on them; thus the total transverse electric force, as observed, 
will be reduced from the above value in the ratio (% — Vj) / + t’^), where % and tq 
are the velocities of drift of the positive and negative ions, which may be different 
just as Kohlrausch found them to be in ordinary electrolysis. The absolute 
velocity V of an ion does not affect the result in this case. This view would 
therefore make the sign of the Hall effect depend on whether positive or negative 
ions conveyed most of the current. 
120. The electromagnetic or mechanical forces acting on the conductors conveying 
the currents are on the other hand to be derived from the energy function, considered 
as potential after change of sign as in § 57, by the method of variations. For the 
reasons given above, the effect of the term involving intrinsic electric inertia, 
is in the present problem inappreciable, except as giving a kind of internal gaseous 
pressure if the velocities of free electrons were comparable with that of radiation. 
The total electrokinetic energy is thus practically 
Ij’Mtc/sTds', where M = cos [ds, ds) + \d~r j ds dd ; 
and on the present hypothesis the energy may be considered to be correctly localized 
in this formula. 
If the currents are uniform all along the linear conductors, the second term in M 
integrates to nothing when the circuits are complete, and we are thus left with the 
Ampere-Neumann expression for the total energy of the complete currents, from 
which the Amperean law^ of force may be derived in the known manner by the 
method of variations. But it must be observed that, as the localization of the energy 
is in that process neglected, the legitimate result is that the forcive of Ampere, 
* It is assumed here that all forces of electric origin acting on the moving atomic charges are 
primarily electric forces; in accordance with the previous theoi'y (§ 57) it is only the part of the 
energy-charge which cannot be compensated by electromotive work, that reveals itself ultimately as a 
forcive working mechanically on the aggregates which constitute conducting bodies, or as heat in case 
it is too fortuitously constituted to admit of transformation into a regular mechanical working forcive. 
This ultimate destiny is independent of any question as to the origin of the inertia of the atoms. Thus 
the steady and unlimited fall of the electric resistance of metals with lowering of the temjieraiure, 
found by Dewar and Fleming, shows that the frittering away of electric energy into heat in a metallic 
conductor depends upon the velocity of fortuitous agitation of the molecules, and would disappear when 
it ceased. The regular transfer of the electrons would thus involve no degradation of electric energy 
(§ 115), except so far as it is disturbed and mixed up by the thermal agitations of the molecules of the 
conductors. In electrolytes the dependence of the degree of ionisation on the temperature may mask 
the cl irect effect of the thermal agitations. 
