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Proceedings of Royal Society of Edinburgh. [sess. 
considered, but on a conductor such as the inductor of a dynamo of 
any kind, momentarily in circuit with it, is only skin deep if the 
duration of the electromotive force be but short enough ; and that 
the depth to which the current penetrates in a given very short 
time is much smaller for iron than for copper. This is certainly the 
explanation of Lord Armstrong’s wonderful experiment. 
To find something towards a mathematical solution for the in- 
creasing current at any instant during the electric contact of Lord 
Armstrong’s experiment, it is convenient first to solve the problem 
of finding the subsidence of current initially given in a circuit of 
two very long parallel bars connected by end bridges, or in a circuit 
of one long bar insulated within a conducting sheath except at its 
ends, which are in metallic connection with the sheath. 
In all cases of electric currents given in parallel straight lines, 
and left to subside,* without any other electromotive force than that 
of their mutual electro-magnetic induction, the thermal analogy is 
exceedingly convenient. For electric conductivity, c, we have 
thermal capacity divided by 47r; for magnetic permeability, the 
reciprocal of thermal conductivity ; and for current-density, tempera- 
ture multiplied by thermal capacity. Thus, if two infinitely long 
straight parallel conducting bars, separated by insulating material, 
be given with equal currents in opposite directions through them, 
and left to themselves, we have precisely the same mathematical 
problem to solve as if in every line of the thermal analogue, we had 
* The thermal analogue for a varying or constant electromotive force applied 
by a voltaic battery or dynamo, substituted for one of the end-bridges, is 
positive and negative sources of heat applied at the interfaces between the 
thermal analogues of electric conductor and electric insulator. The quantity 
of heat generated per unit of area per unit of time, at any point of either 
interface in the thermal analogue, is equal to the rate of variation per unit of 
length along the electric conductor, of the electrostatic force in the insulator 
in contact with it in the electric analogue. Remark that, in the electric j 
system the potential is uniform over each normal section of either conductor, 
and that the variation of potential within each conductor per unit of distance 
along its length — that is to say, the component electrostatic force in the 
direction of the length is exceedingly small in comparison with the component 'i 
electrostatic force perpendicular to the length at any place in the insulator, 
except close to the ends metallically connected by a bridge. The equations in 
the text are unchanged, except the second interfacial condition, it becomes 
where a denotes the quantity of heat generated per unit of time in the source. 
