PEOEESSOE CATLET ON PEEPOTENTIALS. 
763 
thus introducing for the moment s+2 variables X, Y, . . . Z, W, T, which satisfy iden- 
tically X 2 +Y 2 ... +Z 2 +W 2 — T 2 =0, then considering these as functions of the fore- 
going s independent variables 6, <p, ... ip, we have 
d S= 
l 
T s+1 
X, Y ... Z, W 
dX dY dZ dW 
dd’ d&"‘ dQ 
dd dtp ...dip= 
1 d(X,Y...Z, W) 
T 1 B (3, p ■ • • *) 
dd dtp ... d-\r 
dX dY dZ d W 
d<p ’ d<p ' ’ ' d<p’ d<p 
dX dY dZ dW 
dif /’ dv[/ " d\J/ d\f/ 
141. Considering next the s+2 variables X, Y, . . . Z, W, T as linear functions (with 
constant terms) of the s+1 new variables or say as linear functions of the 
s-}-2 quantities «y, 1, which implies between them a linear relation 
«X + 6Y . . . +cZ+dW + eT=l ; 
and assuming that we have identically 
X 2 +Y\ . . +Z 2 +W 2 -T 2 =r-H 2 • • • +r+* 2 -l, 
so that in consequence of the left-hand side being =0, the right-hand side is also =0; 
viz. j] a are connected by 
i 2 +;j 2 ...+r+" 2 =l: 
let dX represent the spherical element belonging to the coordinates a. Con- 
sidering these as functions of the foregoing ^independent variables Q, cp, . . . ■ty, we have 
d%= 
7) . . 
.. £ 
CO 
d$ 
dr, 
. d X 
dw 
dd’ 
dQ 
dw’ 
ud 
d£ 
dr, 
% 
dw 
dp? 
dp 
dp’ 
dp 
dk 
dr, 
dco 
dp’ 
dip 
. . 
dip’ 
dip 
142. We have in this expression u, each of them a linear function of the 
s+2 quantities X, Y, . . . Z, W, T ; the determinant is consequently a linear function of 
s -j- 2 like determinants obtained by substituting for the variables any s-J- 1 out of the s-j-2 
variables X, Y . . . Z, W, T ; but in virtue of the equation X 2 -f-Y 2 . . . + Z 2 -f W 2 — T 2 =0, 
mdccclxxv. 5 i 
