where 
of Edinburgh, Session 1873-74. 227 
zrcL^ = — Sa x V. \f/a , 
and it is easily seen that 
, ra 
^=5V- 
13. In the case first treated, the couple exerted by one current- 
element on another is (§ 8 above) 
V. a/srC^ , 
where, of course, ± wa x are the vector forces applied at either end 
of a. Hence the work done when a changes its direction is 
— S.SaVc^ , 
with the condition 
S.a'Sa' = 0 . 
So far, therefore, as change of direction of a alone is concerned, 
the mutual potential energy of the two elements is of the form 
S.a^Oj . 
This gives, by the expression for w in § 8, the following value 
PSa'c^ + QSaa/SaaL . 
Hence, integrating round the circuit of which a x is an element, we 
have ( On Green's and other Allied Theorems , § 11, Trans. H.S.E., 
1869-70) 
y^PSactj -f- QSaa'Sac^) = ff isfi . U V (P a -f- QaSaa') , 
=//d, l s.Vv 1 - «- oq ) , 
= ffds 1 $.TJv 1 Vaa<& , 
where 
* = Ta +Q - 
Integrating this round the other circuit we have for the mutual 
potential energy of the two, so far as it depends on the expression 
above, the value 
ffdsfi.UvJYaa'Q 
= - ffdsfi.VvJfds'V.YiVv'vy® 
= ff ds Jf d »' •[ S.U n W(2$ + Tatf) + SaWSaUv, ?L J . 
