1891.] On the Theory of Electrodynamics. 525 



elements is now supposed to be specified by the elements of this 

 integral. 



The form of ^ is limited by the fact that it must be a function of 

 the geometrical conformation of the pair of elements. The elements 

 of this conformation are given by the equations 



dr 



cos 6 = , 



ds 



n i dr 



COS 9 = ;, 



as 



COS 



d ( dr\, 

 1 r I 



ds\ ds) 



. dr dr d^r 



ds ds' ds ds' 



where r is their distance apart, 0, #', -e represent the angles ) . ds, r . ds', 

 ds.ds', r being measured positive from ds to ds'. 



The only function of the type dt^lds ds' which can be specified in 

 terms of these quantities is d 2 0(r)/ds ds', which is equal to 



r -1 0' (r) (cos 6 cos 0' cos e) +0" (r) cos cos & '. 

 On substitution we have 



T= 



r ds ds' ds ds 



in which the elements of the energy are supposed to be correctly 

 localised. 



To obtain the mutual mechanical forces between the conductors we 

 have to determine the variation in T produced by the most general 

 virtual displacements of the separate elements which do not alter 

 these elements, nor break the continuity of either circuit. Thus 

 Is, ds', i, i are not to be varied. 



The shortest way to take, account of currents which are not of the 

 same strength all along the circuit is to consider two uniform currents 



i flowing in interrupted circuits, and examine the terms of the 

 variation involving the terminal points at which electric charges are 



3i'ng accumulated by the currents flowing into them. Of course 

 the same general results would flow from taking t, i functions of s, s' 



3spectively and neglecting the ends. Thus, employing electro- 



lagnetic units and so avoiding a numerical coefficient, we have, 



fter F. E. Neumann and von Helmholtz, 



2 N 2 



