346 
A MEMOIR ON PREPOTENTIALS. 
[607 
But in this equation the functions W' and W" each of them belong to a point 
(x,.., z, w) on the surface, and we have at the surface W' — W", = W suppose; the 
term on the right-hand side thus is J W 4- dS, which vanishes in virtue of 
— 0 ; and the equation thus becomes 
as aa 
dW dW’ 
da + da' 
dS = 
4 (r 
that is, the point (a,.., c, e) being interior, we have 
'-T(\s-%) (dW dW"\ dS 
V': 
F' = /' 
4(r*)> 
da' da” ) {(a — xf + ... + (c — zf + {e — w) 2 p * 
In exactly the same way, if (a,.., c, e) be an exterior point, then we have 
U^r dS= i W'~dS, 
da J da 
u 
dW' 
da" 
dS= I w 
dU 
da" 
dS- 
4 (ri) s+1 
r (*«-» 
V"; 
adding, and omitting the terms which vanish, 
\ u 
that is, 
r „ r-TQs-i)/dW’ ■ 
7 da 
dW dW h 
da' + da" 
(dW' dW'_ 
id W 
\ da' 
4 (r^) s+1 
42. Comparing the two results with 
V = 
r ( 2 «— 2) 
dS 
{(a — xf + ... -f (c — zf + (e — w) 2 p * ’ 
pdS 
{(a — xf + ... + (c — zf + (e — iy) 2 }* s- i ’ 
we see that, V' and V" satisfying the foregoing conditions, there exists a distribution p 
on the surface, producing the potentials V' and V" at an interior point and an 
exterior point respectively; the value of p in fact being 
T(\s-\)(dW’ , dW 
•(C), 
4 (T^f +1 v da' + da" 
where W\ W" are respectively the same functions of (x,, z, w) that V', V" are of 
e). 
The Potential-solid Theorem D. Art. No. 43. 
43. We have as before (No. 40), 
\w~dS + f dx... dz dw FV U - V 
J da J r (is - A) 
(V 
dW 
da 
r <*«-*) 
dS + I dx ... dz dw i7V W — 
,S+1 
4 (r^> 
r(^-i) 
T,