PEOFESSOE CATLET ON PEEPOTENTIALS. 
703 
tinuous. Taking, then, for the closed surface S the boundary of infinite space, U and 
W each vanish at this boundary, and the equation becomes 
-( r i)l^V= Ux... dz dw UVW ; 
r(i«— i) •/ 
viz. substituting for U its value, and comparing with 
g dx . . . dz dw 
{(«— xy . . .+(c— zy+(e— wyy 
where the integral in the first instance extends to the whole of infinite space, but the 
limits may be ultimately restricted by g being =0, we see that the value of g is 
W being the same function of (x ... z, w) that V is of (a ... c, e), which is the theorem D. 
Examples of the foregoing Theorems. — Art. Nos. 44 to 49. 
44. It will be remarked, as regards all the theorems, that we do not start with known 
limits ; we start with V a function of (a ... c, e), the coordinates of the attracted point, 
satisfying certain prescribed conditions, and we thence find g, a function of the coordinates 
(x...z) or (x...z,w), as the case may be, which function is found to be =0 for 
values of (x . . . z) or (x ... z, w) lying beyond certain limits, and to have a determinate 
non-evanescent value for values of (x . . . z) or (x ... z,w) lying within these limits ; and 
we thus, as a result, obtain these limits for the limits of the multiple integral V. 
45. Thus in theorem A, in the example where the limiting equation is ultimately 
found to be x 2 . . . -f -z 2 =f 2 , we start with V a certain function of a 2 . . . -f -c 2 {=%? suppose) 
and e 2 , viz. Y is a function of these quantities through 9, which denotes the positive root 
of the equation 
the value in fact being V=j t~ q ~ l (t-\-f 2 )~ is dt, and the resulting value of g is found to 
be =0 for values of (x ... z) for which x 2 . . . +z 2 >/ 2 . Hence V denotes an integral 
J {{a— a?) 2 ... + (c— ^) 2 + e 2 }^ +2 ’ 
the limiting equation being x 2 . . .-\-z 2 =f 2 , say this is the s-co ordinal sphere. 
And similarly, in the examples where the limiting equation is ultimately found to be 
x 2 - z 2 
j 2 ... + ^ 2 =l, we start with Y a certain function of #, ...c, e through 6 (or directly 
and through 0), where 0 denotes the positive root of the equation 
