768 
MR. A. McAULAY ON THE MATHEMATICAL 
This force cannot apparently be explained by any term in l of the kind we have 
hitherto supposed l to contain. Suppose, then, l to contain a term # SdaA where a is 
of the same class as A or ur [equations (5), § 81.] Thus ctcr is an intensity, and aj a 
flux, a c being the conjugate of a . 
The portion of L contributed by this term is JjjSdaAc/?, or [eq. (4), § 5], jJSdac?2. 
Hence the effects of supposing l to contain a term SdoA are identical with those of 
supposing l s to contain a term [SdrtUV] a + &. 
So far as electric phenomena are concerned it is quite easy to see the eflect of this 
term. In place of equations (31), § 50, and (2), § 57, we shall have 
E, s . = [rU v] a + h = [aU v\ a + h .(2), 
or, if a be a scalar, 
= .( 3 )- 
100. Although this term involves a contact force it does not explain the known 
facts, since, as can be easily seen, the contact force here obtained is such that 
equation (1) would be invariably true. We seem then to be driven to the conclusion 
that to explain contact force, l s cannot any longer be assumed to be zero. 
Adopting the suggestion of Professor Chrystal, £ Encyc. Brit.,’ 9th ed., vol. 8, p. 85, 
we will assume that there is no real contact force between conductors. This simplifi¬ 
cation is not, of course, necessary on the present theory, but the simpler the 
assumptions—-so long as they are not intrinsically improbable—the better. Professor 
Chrystal shows that the assumption serves to explain all the known facts, the 
apparent contact force between conductors being explained by the difference of their 
contact forces with one and the same dielectric. 
We now assume that l s contains the termt SaUV c [d] a + i/2 where a is of the same 
class as before, but now depends on ^ and 6, and where of course the suffix a has 
nothing to do with the linear vector function a. It is assumed that a is zero for a 
surface of separation of two conductors. 
In place of equations (2), (3), we now have 
E s — [_vJJv] a + b = a\Jv o; .(4) 
W.-1 = «.(5). 
* It may be asked, Why make the differentiations act on a as well as d, why not assume the term to 
he Sd 1 aV 1 ? The answer is that this leads to a more complicated result, namely (1), to the same contact 
force as the term chosen, (2), to a term — aA in E, and (3), to a stress which involves the space deriva¬ 
tives of d. It is best to assume, if possible, a term which involves what we know to be true and 
nothing more. 
f Perhaps it would be better, as more g’eneral to suppose l, to contain the term [SMJV d] a + j where ot. 
is of the same class as a, and ah is not merely characteristic of the substance b, but depends on both the 
substances hounded, i.e., where, in general, j + a/,_ c + is not zero. Equations (4), (5), (6) will 
still be true if we put a — 
