1889-90.] Sir William Thomson on Ohmic Resistance. 
161 
initial temperature multiplied by thermal capacity given equal to 
the current-density in the corresponding line of the electro-magnetic 
problem, and the system left to itself, with the positive and negative 
temperatures in the two bars subsiding towards zero. 
The thermal analogue for the insulating material of the electro- 
magnetic problem is an ideal medium of zero thermal capacity. 
Thus in process of equalisation of temperature we have diffusion of 
heat through the substance of each bar, according to Fourier’s 
original use of the term diffusion ; while in the ideal medium taking 
the place of the electric insulator, we have merely conduction of 
heat, without any diffusion properly so called, that is to say, without 
any excess of heat conducted out of, above heat conducted into, any 
portion of the medium. 
If £ denote the temperature at time t , in the thermal analogue, 
at any point, P ; 47 re the thermal capacity per unit of volume ; 1 /W 
the thermal conductivity of the analogue to either of the electric 
conducting bars ; and 1 /z«r' the thermal conductivity of the analogue 
to the insulating medium: the equations expressing all the conditions 
of the problem are 
4 m=±(^ + fi), 
dt \dx 2 dy 2 J 
[P, in either bar] 
dx 2 dy 2 ’ 
[P, in the analogue to the insulating medium] : 
RMCT 
im = im 
tAjA \jra 
■ [P, in the interface] ; 
where djdv denotes rate of variation per unit length in the direction 
of the normal, at any point of the interface ; and [ ], and [ ]' denote 
values infinitely near the interface outside and inside respectively. 
In the particular case of one of the bars of circular cross-section, 
and the other a hollow circular cylinder surrounding it coaxially, 
the problem becomes greatly simplified. It becomes still farther so 
if we suppose the electric conductivity, or in the thermal analogue 
the thermal capacity, of this outside sheath to be infinitely great. 
In this last case we have identically the same mathematical problem 
VOL. xvii. 4/7/90 
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