PROEESSOK CAYLEY ON PREPOTENTIALS. 
677 
It is, in fact, only the prepotential-plane case which is connected with the partial 
differential equation 
(&_ d?_ 
\flfa 2 * ’ ‘ ^ dc 2 ' 
de 2 ' e 
Y=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 
l 
| a 2... + C 2 + e 2}i*+?’ 
and therefore also hy 
l 
{(a-^) 2 ... + (c-z ) 2 + e 2 } is+9 ’ 
and consequently by the integral 
f g dx . . . dz (A) 
J {(a-x)‘ 2 ... + {c-z ] 2 + e 2 }> s+q 
that is hy the prepotential-plane integral ; but the equation is not satisfied by the value 
{{a—xf . . .+ (c— z ) 2 + (e — w) 2 } is+g 
nor, therefore, hy the prepotential-solid, or general superficial, integral. 
But if ^=— 1, then, instead of the prepotential equation, we have “the potential 
equation ” 
(* + *.£W_ 0 . 
and this is satisfied by 
and therefore also by 
Hence it is satisfied by 
{a 2 . . . + c 2 + e 2 } 2 *’" 2 
1 
{(a— x ) 2 . . . + (c— z) 2 + (e— m;) 2 } 2 * 2 
dz dw 
C qdx . . 
J Ua— x ] 2 . . . + (c- 
{(a— x ) 2 . . . + (c— z ) 2 + (e— w) 2 } is 2 ’ 
(P) 
the potential-solid integral, provided that the point (a ... c, e) does not lie within the 
material space : I would rather say that the integral does not satisfy the equation, but 
of this more hereafter ; and it is satisfied by 
f ; rn 
J {(a — x ) 2 . . . + (c— z ) 2 + (e— w) 2 } 2 * 2 ’ ' ' 
the potential-surface integral. The potential-plane integral (B), as a particular case of 
(C), of course also satisfies the equation. 
Each of the four cases give rise to what may be called a distribution-theorem ; viz. 
given Y a function of (a . . . c, e ) satisfying certain prescribed conditions, but otherwise 
arbitrary, then the form of the theorem is that there exists and that we can find an expres- 
4x2 
