760 
PBOEESSOK CAYLEY ON PBEPOTENTIALS, 
("pns+i 
Assuming for M the value x|y/* +1 , the l ast value becomes § = 1 ; and writing for 
shortness ar . . . +c 2 +e 2 =;s 2 , we have 
_ (IAp+> .f+i 
, for exterior point z>f, 
worked out by the theorem ; this is in fact what is done in tridimensional space by 
Lejeune-Diriciilet in his Memoir of 1846 above referred to. 
Annex VIII. Prepotentials of the Iiomaloids. — Nos. 135 to 137. 
135. We have in tridimensional space the series of figures — the plane, the line, the 
point; and there is in like manner in (s-j-l)dimensional space a corresponding series 
of (s + 1) terms; the (sfi-l)coordinal plane — the line, the point: say these are the 
homaloids or homaloidal figures. And (taking the density as uniform, or, what is the 
same thing, =1) we may consider the prepotentials of these several figures in regard to 
an attracted point, which, for greater simplicity, is taken not to be on the figure. 
136. The integral may be written 
which still relates to a (s-fl)dimensional space: the (s+1) coordinates of the attracted 
point instead of being (a . . . c, e) are (a. .. c, d ... e,u) ; viz. we have the s' coordinates 
(a . . . c ), the s— s' coordinates (d. . . e), and the (s+l)th coordinate u : and the integration 
is extended over the (s—s') dimensional figure w~ — co to -f-oo,...£= — oo to +oo . 
And it is also assumed that q is positive. 
It is at once clear that we may reduce the integral to 
v =^ 
{ (a-a?) 2 ...+ (c— ^) 2 + M 2 H-^y 2 ... + ^ 2 }' s+?, 
dw ... dt 
say for shortness 
dw ... dt 
(A 2 + w 2 ... + * 2 ) ii+? ’ 
where A 2 , ={a—x'f... J r {c—zf-\-u 2 , is a constant as regards the integration, and where 
the limits in regard to each of the s — s' variables are — oo , -|-oo . 
We may for these variables write . . .r£, where | 2 . ..-1-^ 2 =1 ; and we then have 
