TRANSACTIONS OF SECTION A. 733 
ohmic resistances are so moderate! that the quantities of electricity deposited on 
and removed from their boundaries to supply the electrostatic forces along the 
conductors required for producing the alternations of the currents, are negligeable 
in comparison with the total quantity flowing in either direction in the half period. 
This supposition excludes important practical problems of telegraphy and tele- 
phony, the problem of long submarine cables, for instance; but it includes the 
problem of electric lighting by alternating currents transmitted at high tension 
through considerable distances; as, for example, from Deptford to London, 
4, The general investigation includes as readily any number of separate circuits 
of parallel conductors as a single circuit, but, for simplicity in describing results, I 
suppose our system of conductors to be so joined at their ends as to constitute a 
single simple circuit of two parallel conductors? It may be either two parallel 
conductors or one conductor, one of which may or may not surround the other, as 
shown in Figs. 1 and 2, representing cross-sections. Each conductor may be 
single, as in Figs. 1 and 2, or either may be multiple parallels. 
Fig. 1. 
5. We suppose each conductor to be homogeneous in substance, and in cross- 
section from end to end, but not necessarily homogeneous in different parts of the 
cross-section. Thus the two conductors, or the different parts of either, may be 
of different metals, or either conductor or any part of either conductor may consist 
of me metals (as iron and copper, or iron and lead) laid parallel and soldered 
together. 
an We shall call A and A’ the cross-sectional areas or groups of areas of the 
two conductors respectively of the other. All the different portions of A are con- 
nected metallically at their two ends, and are thus all of them at one potential at 
one end and another potential at the other end; and similarly for A’. The homo- 
geneousness of the material and of the cross-sections along the length of the con- 
ductors and the uniformity of the total currents assumed in section 3, implies that all 
the different parts of A in one cross-sectional plane are at one potential, even 
though A consist of mutually isolated parts, or A’ consist of isolated parts. If, as 
1 The circumstances in which this condition is fulfilled may be usefully illustrated 
by considering the important practical cases of submarine cables, and of metallic 
circuits of two parallel wires insulated at a distance anything less than a few 
hundred times their diameter. For all these cases the numeric expressing the 
electrostatic capacity of either conductor per unit length (the other supposed for the 
moment to be at zero potential) is between 2 and 0:1, and for our present rough com- 
parison may be regarded as moderate in comparison with unity. On this supposition 
the condition of the text requires for fulfilment that the mean proportional between 
the velocity which expresses in electro-magnetic measure the resistance of one of the 
conductors and the velocity of a body travelling the length of the conductor in a time 
equal to half the period of alternation, shall be exceedingly small in comparison 
with the velocity of light. 
? The case of a single circuit made up of parallel conductors, so joined at their 
ends that to travel once round it we must go and come two or three or more times 
along separate conductors, joined by their ends in series, so as to make one circuit, 
is specially considered in my Paper on ‘ Anti-Effective Copper in Parallel or in Coiled 
Conductors for Alternating Currents,’ p. 736. 
