320 
A MEMOIR ON PREPOTENTIALS. 
[607 
C. The potential-surface case; q = — the surface arbitrary, so that the integral is 
i p dS 
J {(a - xf + ... + (c - zf + (e - w) 2 p-5' 
D. The potential-solid case; q = — ^, and the integral is 
p dx ... dz dw 
f {(a - xf 
+ ... + (c — zf + {e — wf)* 8 '* ‘ 
It is, in fact, only the prepotential-plane case which is connected with the partial 
differential equation 
d- d- d 2 2o +1 d 
— -i_ ... _1 ^ -—. _|—•* 
da 2 dc 2 de 2 e de 
V=0, 
considered in Green’s memoir * “ On the Attractions of Ellipsoids ” (1835), and called 
here “the prepotential equation.” For this equation is satisfied by the function 
and therefore also by 
{or + ... + c 2 + e 2 }* s+q ’ 
1 
{(a — x) 2 +... + (c — z) 2 + e 2 p+9 ’ 
and consequently by the integral 
pdx... dz 
f P± 
J {(a-x) 2 +... 
■(A), 
+ (c - z) 2 + e 2 )^ s+q 
that is, by the prepotential-plane integral; but the equation is not satisfied by the value 
1 
{(a — x) 2 + ... + (c — zf + (e — wf)* s + q ’ 
nor, therefore, by the prepotential-solid, or general superficial, integral. 
But if q = — \, then, instead of the prepotential equation, we have “ the potential 
equation ” 
d 2 dr d 2 \ 
+ ••• + + 
and this is satisfied by 
and therefore also by 
Hence it is satisfied by 
. v= 0 • 
da 2 de 2 de 2 ) ’ 
[a 2 + ... + c 2 + e 2 f s ~* * 
{(a - xf + ... + (c — zf + (e — w) 2 p-* ’ 
I {(a —x 
pdx ... dz dw 
{{a — xf + ... + (c — zf + (e — w) 2 }* 8- - 
•(D), 
* [Green's Mathematical Papers, pp. 185—222.]