222 
Proceedings of the Royal Society 
involve a only. That this is the case follows from the fact that 
pa' is homogeneous and linear in each of a v a'. It farther follows, 
from IV., that the part of pa' which does not disappear after 
integration round each of the closed circuits is of no dimensions 
in Ta, Ta, Ta x . Hence ^ is of - 2 dimensions in Ta, and thus 
_ paSaa, qa } rYaa x 
X 1 Ta 4 Ta 2 + Ta 3 
where p, q, r are numbers. 
Hence we have 
_ / q/ ^ 7 \ » / . pVaa'Saa. qYa'a. 
pa' = - S(a 1 V)'K + Ta4 — t + 
rY.a'Y aa x 
+ Ta? 
Change the sign of a in this, and interchange a' and a x , and we 
get the action of a on a x . This, with a and a x again interchanged, 
and the sign of the whole changed, should reproduce the original 
expression — since the effect depends on the relative, not the abso- 
lute, positions of a, a 1? a. This gives at once, 
and 
P = o , q = 0, 
rY.a'Y aa. 
pa' = — S(a 1 V)i^a / + • 
Ta 3 
with the condition that the first term changes its sign with a, and 
thus that 
pa' = aSaa'F(Ta) 4 - a'F(Ta), 
which, by change of F, may be written 
= aS(a'V)/(Ta) + a'F(Ta) , 
where / and F are any scalar functions whatever. 
Hence 
pa' = - S(a 1 V)[aS(a'V)/(Ta) + a'F(Ta)] + 
rY.a'Yaa x 
Ta 3 
which is the general expression required. 
3. The simplest possible form for the action of one current- 
element on another is, therefore, 
pa 
rY.a'Yaa x 
TW 3 
