418 
A MEMOIR ON PREPOTENTIALS. 
[607 
the surface and = 0 for points outside the surface; and then in the potential or 
prepotential integral they extend the integration over the whole of infinite space, thus 
getting rid of the equation of the surface as a limiting equation for the multiple 
integral. 
152. Lejeune-Dirichlet’s paper “Sur une nouvelle méthode pour la determination 
des intégrales multiples” is published in Gomptes Rendas, t. viii. pp. 155—160 (1839), 
and Liouville, t. iv. pp. 164—168 (same year). The process is applied to the form 
d 
dx dy dz 
p — 1 da J ((a — acf + (6 — yf + (c — z) 2 }i<i > - 1 > 
/^2 ^#2 
taken over the ellipsoid — + ~- 2 + — = 1; but it would be equally applicable to the 
CL ¡D y 
triple integral itself, or say to the s-tuple integral 
f dx ...dz 
or, indeed, to 
j \(a- 
{(a — x) 2 + ... + (c — ,z) 2 p’ + 3 ’ 
dx ...dz 
{(a — x) 2 + ... + (c — z) 2 + e 2 }* 8+ ? 
a? 
taken over the ellipsoid y^+...+p=l; but it may be as well to attend to the first 
form, as more resembling that considered by the author. 
153. Since — f cos \cf> d(p is =1 or 0, according as X, is <1 or > 1, it 
Jo 
tt Jo </> 
follows that the integral is equal to the real part of the following expression, 
d(f> 
sin (f) [ <(J-2+-+S) 
dx... dz 
<f> J 
{(a-xy+...+(c- z) 2 )^ s+ ^’ 
where the integrations in regard to x,..,z are now to be extended from — oo to + oo 
for each variable. A further transformation is necessary: since 
1 l . f 00 
— = |Y e rn J d\fr.yjr r 1 e l<T ^, <t positive, and r positive and <1, 
writing herein (a — xf+ ... + (c — z) 2 for er, and \s + q for r, we have 
1 1 
{(a — xf + ... 4- (c — zy\* s+( i r(^s+g) 
and the value is thus 
2 
e —(is+q)tri I ' yjfis+q-1 { (a-x) 2+ ... + (c-z)2} ; 
ttT ($s+q) 
e~ (is+q) Y 
1 [ dé [ dyjr . y]ri*+9~' [ e t '(^+-+y («-*)»+«.+(«-*)*} dx... dz, 
Jo 9 J o J