PROFESSOR CAYLEY ON PREPOTENTIALS. 
753 
and assume that this is satisfied by Q=H§zd0, then we have 
and therefore 
"(Sf j , «M+a“*+HS) 
+2 Hs ) 
=0 j 
(8«f + 2PH),+4flH|=0; 
viz., multiplying by 5, this is 
or 
^(H’*)+ipH>*=0, 
viz. substituting for P its value, this is 
Wz Te + h ( 2 #+ : 2 ' • • • + W+e) =l °* 
H fc— 03-2 
V/ 2 + 
and 
Hence, integrating, 
©=CH 
v^+o.^+fl. ..**+*’ 
Q-v-'dO 
C an arbitrary constant, 
X arbitrary, 
H 2 V/ 2 + 0./ + 0...# J + 0’ 
where the constants of integration are C, X ; or, what is the same thing, taking T the 
same function of t that H is of 6 (viz. T is what <p becomes on writing therein 
V/ 2 +* 
y .y - + 1 
^h^ + t 
s/P+tf- • - + ^ 2 V/ 2 +/ . . . + X 2 V/ 2 +/...+A 2 ’ 
in place of a, j3, . . . y respectively), then 
_nn t q l dt 
}e T 2 + t .'gWt TJp+i* 
where x may be taken =co : we thus have 
T7 ^ htt f" 
V=@<p = — CH(pJ 
Recollecting that 
T 2 X /f2 + t.C,Z + t...h? + t 
so that for 0=co we have a 2 +b 2 . . . +c 2 +e 2 =^, the assumption x =co comes to making 
V vanish for infinite values of (a, b,...c, e ). 
