710 
MR. A. McAULAY ON THE MATHEMATICAL 
of such equations of condition is well known. In the present case it takes the 
following form : to the left of equation (24) add 
- | I | Sa (SC - YV SH/ 477 ) ch + (47r) -1 j [ Sa, dZ SH 
where a, a, are vectors ; S C and S II may then be regarded as independent. It is to 
be noted that there is but one a* for an element of the bounding surface, i.e., there is 
not one for each region bounded. In our notation this may be expressed by saying 
that [a,] a = [aji. 
Now, by eq. (4), § 5, above, 
SaV SH ds + | 
Hence, since the part contributed to the left of eq. (24) by SH is — JJj S SH H Va; ds, 
we get, by equating to zero the coefficient of the arbitrary vector SH, 
j Sa, dZ SH = f (| S SHVa ds + jj S (a, + a) dZ SH. 
4tt h Vx = VVa = b.(26) 
[VUvaJ^s = 0 .(27), 
a, disappearing, since [aj fl = [ ajj. 
Again, before considering what is contributed to the left of eq. (24) by SC, it must 
be remembered that SC is not quite arbitrary, by reason of the equations of condition 
SVC = 0, [SU^C] cl + 5 = 0. This is taken account of by adding to the left of # 
eq. (24) 
[[[ YSV SC c7 s + f f Y,S dZ SC = - fff S SCVY ds + [[ (Y + Y,) S dZ SC 
* It may be objected that these equations of condition have already been taken account of in the 
treatment accorded to the more general equations of 4~C = VVH, [VUrH] a+ s = 0, and, therefore, it is 
erroneous to take account of them again. The answer to this is that it is not necessary to do this, but, on 
the other hand, it is not erroneous. We must expect as the result that the Y’s will be, in a mathematical 
sense, redundant. That this actually is the case will appear in § 65 below. The reason for introducing 
them is to obtain the equations of the field in as familiar a form as possible, and to show the mathe¬ 
matical dependence of the existence of a potential on the equations SVC = 0, [SITrCjn+j = 0. The 
process may be paralleled in the subject of the Calculus of Variations. IT, V, W being three functions 
of x, y, . . . , cx, cy, ... , linear in the latter group, let it be required to satisfy the equation IT = 0 
subject to the equations of condition V = 0, W = 0. The recognised method is to use the single 
equation IT + AV + BW = 0 instead of the three, A and B being functions of x, y, . . . determinable 
by the problem in hand. It would not be erroneous to add to the left of the last equation CW, where 
C was a function of the same kind as A and B. One of the two, B or C, would be mathematically 
redundant, but it might bo convenient to introduce both and give arbitrarily, later on, some method of 
assigning a definite meaning to each. 
