PROFESSOR CAYLEY ON PREPOTENTIALS. 
697 
which, if U, W each satisfy the prepotential equation, becomes 
j>-w^s 
And if we now take the closed surface S to be the surface positive infinity, together 
with the plane e=0, then, provided only U and Y vanish at infinity, for each integral 
the portion belonging to the surface positive infinity vanishes, and there remains only 
the portion belonging to the plane e=0 ; we have therefore 
fe 2 « + ' w ~ dx... dz=^e 2q+1 JJ ~ dx . . . dz, 
where the functions U, W have each of them the value belonging to the plane e=0 , 
viz. in U, W considered as given functions of (x ... z, e) we regard e as a positive quan- 
tity ultimately put =0, and where the integrations extend each of them over the whole 
infinite plane. 
34. Assume 
TT 1 
U_ {(«-#... + (c-*) 2 H he 8 } 1 -*-*’ 
an expression which, regarded as a function of (x ... z, e), satisfies the prepotential 
equation in regard to these variables, and which vanishes at infinity when all or any 
of these coordinates (x ... z, e) are infinite. 
We have 
dU — 2{\s-\-q)e 
{(a-xf . . . + (c -*) 2 + e 2 } is+2+1 ’ 
and we have consequently 
=j>«u^<zs. 
w 
-2 {\s+q)e iq ^ 
{(a-xY... + {c-zY + e*} 
1 2ti s+ ®~ 1 
dx ...dz 
=f(-£. 
dx ... dz 
){(«—«) 2 . .. + (c— z ) 2 + e 2 } i 
where it will be recollected that e is ultimately =0 ; to mark this we may for W 
write W 0 . 
Attend to the left-hand side ; take V 0 the same function of a ... c, e=0, that W 0 is of 
x ... z, e=0 ; then first writing the expression in the form 
^ r — 2 (-^s + q) e^i^dx . ..dz 
°J {{a—xf . . .+ (c-.z ) 2 + e 2 }* s+9+I ’ 
write x=a-}-e<' . . . z=c-\-e%, the expression becomes 
— V f ~ 2 (a* + g) e 2g+2 • e s d£ • • • dt 
°J W 
i (i+r...+§*)} i * + * +i 
= -2(i 5 + 2 )V 0 j jI 
+ | 2 ... + ? 2 } 5W 
where the integral is to be taken from — oo to + go for each of the new variables !;...£. 
Writing f=m . . . £=ry, where a 2 . . .-|-y 2 =l, we have -d\ . . . d%=r s ~ 1 dr dS also 
£ 2 . . . + £ 2 =r 2 , and the integral is 
r s ~'dr 
(l+r 2 f s+9+1 ’ 
r s ~ 1 dr 
(l+r 2 f s+9+1 ’ 
