607] 
and observing that 
A MEMOIR ON PREPOTENTIALS. 
413 
the integral becomes 
/ 
where X, Y,.., Z, W, T denote given linear functions (with constant coefficients) of the 
s +1 variables y,, Ç, co, or, what is the same thing, given linear functions of the 
5 + 2 quantities £, y,.., Ç, co, 1, such that identically 
X 2 + Y 2 +... + Z 2 + W 2 -T 2 = % 2 + v *+ ... + ? +<o s -1. 
We have then £ 2 + y 2 + ... + £ 2 + co 2 — 1 = 0, and dS as the corresponding spherical element. 
144. We may have X, Y,.., Z, W, T such linear functions of y,.., Ç, co, 1 that 
not only 
X 2 + Y 2 + ... + Z 2 + W 2 -T 2 = ? + y 2 + ... + r 2 + a> 2 -l 
as above, but also 
(*QX, F,.., Z, W, T) 2 = Ag 2 + Brj 2 + ... + C^ 2 + Eco 2 — L ; 
this being so, the integral becomes 
J {A? + B V 2 + ... + C? + Eco 2 - Z}*> 
where the 5+2 coefficients A, B,.., G, E, L are given by means of the identity 
-(d + A)(d + B)...(d + C)(6+E)(d+L) 
= Disct. Y,.., Z, W, T) 2 +Û(X 2 +Y 2 +... + Z 2 +W 2 -T 2 )}; 
viz. equating the discriminant to zero, we have an equation in 6, the roots whereof 
are — A, — B,.., — G, — E, — L. 
The integral is 
/ 
{(A — L) % 2 + (B — L) rj 2 + ... + (C — L) Ç 2 + (E — L) a> 2 p’ ’ 
which is of the form 
where I provisionally assume that a, b,.., c, e are all positive. 
145. To transform this, in place of the s + 1 variables £, rj,.., £, co connected by 
£ 2 + y 2 + ... + £ 2 + <w 2 = 1, we introduce the 5+1 variables x, y,.., z, w, such that