PEOEESSOK CAYLEY ON PEEPOTENTIALS. 
745 
there are two cases, 
« 2 
exterior, -, 2 . 
, ,2 
interior, ^ . 
is positive root of 1— ^ . . .— -^=0, 
‘ l J 
. + p<l, Q vanishes, viz. the limits in the integral are oo , 0 ; 
q must be negative, 1 -\-q positive as before, in order that the ^-integral may not be 
infinite in regard to the element £=0. 
It is assumed in the proof that m and 1 -\-q are each of them positive ; but, as appears 
by the second example, the theorem is true for the extreme value m— 0; it does not, 
however, appear that the proof can be extended to include the extreme value q= — 1. 
The formula seems, however, to hold good for values of m , q beyond the foregoing limits ; 
and it would seem that the only necessary conditions are \s-{-q, 1-f m, and 1 -\-q-\-m, 
each of them positive. The theorem is in fact a particular case of the following one, 
proved, Annex X. No. 162, viz. 
J { (a— a?) 2 ... + 
■si *•••* 
[c—zY + e 2 ] 
j S+q 
over the ellipsoid 
where <r denotes : assuming <pu=(l — u) q+r ‘ 
J + 1 h z + t t 
<r)x)dx, 
, we have 
<p(<r + (1 — a)x) = (1 — a) q+m (l — x) q+r> 
and the theorem is thus proved. 
111. Particular cases : 
rZ 
m = 0 
f (l dx...dz 
. \_V„ -Z 2 (Tj)«r(i + g ) rf 
’ J[(a-xY- + (c~zY + e 2 ] is+q ~ T(is + g) KJ " 
h)\ dtt- q -\t+p...t+}f)-K 
Cor. In a somewhat similar manner it may be shown that 
f X 2 £ 2 \® 
( rj)«r(i+ g ) 
P(2 S +?) 
{{a— a?) 2 ...+ (c— ^r ) 2 + e 2 } 
Multiply the first by a and subtract the second, we have 
SIFT,— •(/■••*) V . . «+*»)-*• 
1 — 72 — 12) ifl—x)dx...dz 
{(a—xY... + {c—zY + e 2 Y 
or writing q -{- 1 for q, this is 
Jr P"- h*. 
- T t\ f> • ^^+/ 2 - • • < + *> 
t+r 
f X* 2 2 ,2 \ ?+1 
{ a ~ x ')dx . . dz 
j- q -\t+f\..t+V)-h 
_lL__ (ri)*r(2.+g) (f h) r df 
{ (a-xp.. + (c-zY+e*} is+q+1 T&s + q + l) X t+p 
and we have similar formulae with (instead of (a— x)) . . . c— z, e in the numerator. 
