764 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
these s-j-2 determinants are proportional to the quantities X, Y . . . Z, W, T respec- 
tively, and the determinant thus assumes the form 
«X + b Y . . . + cZ + d W + eT . 
T 
where A is the like determinant with (X, Y, . . . Z, W), and where the coefficients 
a,b, . . . c, d, e are precisely those of the linear relation «X-j-&Y . . . +cZ-|-dW-}-eT=l ; 
the last-mentioned expression is thus =q? A, or, substituting for A its value, we have 
_ 1 d (X, Y . . .Z, W) , ,. 
rp ^(j) a. . . j,*) dQd<p...d\ p; 
d{&, <P- ■ ■4’, 
viz. comparing with the foregoing expression for dS we have 
JS=±dX, 
which is the requisite formula for the transformation of dS. 
148. Consider the integral 
{*Jsc, y . . .z,w, l) 2 }*®’ 
which, from its containing a single quadric function, may be called “ one-quadric.” Then 
effecting the foregoing transformation, 
and observing that 
x,y.. 
X Y 
T’ T’ 
Z W 
T’ T’ 
W,1) S =L(*XX, Y...Z,W,T) 2 , 
the integral becomes 
4 
J)(*XX, Y...Z, W, T)4 
where X, Y . . . Z, W, T denote given linear functions (with constant terms) of the s+1 
variables q . . . £, &>, or, what is the same thing, given linear functions of the s + 2 quan- 
tities g, j? . . . £, 1, such that identically X 2 -)-Y 2 . . . 4-Z 2 ff-W 2 — T 2 =| 2 + ^ 2 . . . -f-£ 2 -|-<y 2 — 1. 
We have then £ 2 -|-jj 2 . . . 4-£ 2 +£y 2 — 1 = 0, and d$ as the corresponding spherical 
element. 
144. We may have X, Y . . . Z, W, T such linear functions of <y, 1 that not 
only 
X 2 +Y 2 . . . + Z 2 -|-W 2 -T 2 =f -H 2 . . . + £ 2 +*, 2 -l 
as above, hut also 
(*XX, Y, . . . Z, W, T) 2 =A£ 2 +1V . . . +C£ 2 + EW 2 -L ; 
