THEORY OF ELECTROMAGNETISM. 
737 
extended round the closed circuit. The quantity z is, therefore, indeterminate as far 
as regards the problem now before us, in which the total electromotive force round 
the circuit is to be determined. We shall find, however, that when we know all the 
circumstances of the problem, we can assign a definite value to 2 , and that it repre¬ 
sents, according to a certain definition, the electric potential at the point p.” Now, 
I have looked in vain through the subsequent part of his treatise to find the promised 
definition of electric potential, and I have tried hard on Maxwell’s own assumptions 
to see how the definite value he here speaks of is to be assigned, and I have totally 
failed. He nowhere shows how to assign a definite value to A'; whereas he certainly 
assigns a definite value to B', and also from equations (9) (10) above, he also clearly 
assigns a definite value to E 0 '. From the equation just given, then, it follows that V'z 
must be indefinite in order to counterbalance the arbitrary part of 0A '/dt, which is 
necessarily of the form V' ( some scalar).* Leaping over this difficulty of Maxwell’s 
assertions, however, i.e., supposing 3A' jdt definite, the question still remains what is 
the definite value of z ? Light seems to be thrown on the question by the assertion 
above that it is the “electric potential,” and the following, taken from § 630 of his 
treatise :— 
“ The energy of the system may be divided into the Potential Energy and the 
Kinetic Energy. 
“ The potential energy, due to electrification, has already been considered in § 85. 
It may be written 
W = A2$>2, 
where T) is the charge of electricity at a place where the electric potential is 2 , and 
the summation is to be extended to every place where there is electrification. 
“If d 7 is the electric displacement, the quantity of electricity in the element of 
volume df is 
2) = - SV'dW, 
and 
W = - a ([[ zSV'dW 
where the integration is to be extended throughout all space.” He then shows that 
it follows that 
W = 1 j|j Sd'V'zdf, 
and proceeds r 
* For VV' A' is assigned for every point of space and [Yd2'A'j« + * ~ 0. It is well known that when 
this much and no more of a vector is known, it contains an arbitrary term V' (a scalar), and that this is 
the full extent of its arbitrariness. 
5 B 
MDCCCXCII.—A. 
