PROFESSOR CAYLEY ON PREPOTENTIALS. 
771 
155. If the point is interior, . . . 4-^< 1, and consequently also a< 1, and the value, 
writing (T-^) 2 instead of 7 r, is 
= r'(is+})r(i-}) (/• • • 0 • (i-Ar-- - A<) • 
^2 ^»2 
But if the point be exterior, -p. • • • + ^> 1, an< ^ lienee, writing 0 for the positive root 
a 2 c 2 
of the equation, <r=l ; viz. 0 is the positive root of the equation • • + ^ 2 qr§—l> ^ ien 
£=0, a is greater than 1, and continues so as t increases, until, for t=6, <r becomes =1, 
and for larger values of t we have <r < 1 ; and the expression thus is 
(nr 
r(Wg)r(i 
h* + t 
dt . . . t+h*)-i (1 
viz. the two expressions in the cases of an interior point and an exterior point respec- 
tively give the value of the integral 
dx...dz 
{{a-x)*...+{c-z)*} is+q 
This is in fact the formula of Annex IV. No. 110, writing therein e=0 and m=—q. 
156. Boole’s researches are contained in two memoirs dated 1846, “On the Analysis 
of Discontinuous Functions,” Trans. Boyal Irish Academy, vol. xxi. (1848), pp. 124-139, 
and “ On a certain Multiple Definite Integral,” do. pp. 140-150 (the particular theorem 
about to be referred to is stated in the postscript of this memoir), and in the memoir 
“ On the Comparison of Transcendents, with certain applications to the theory of 
Definite Integrals,” Phil. Trans, vol. 147, for 1857, pp. 745-803, the theorem being the 
third example, p. 794. The method is similar to that of, and was in fact suggested by, 
Lejeune-Dirichlet ; the auxiliary theorem made use of in the memoir of 1857 for the 
representation of the discontinuity being 
f(x) 
t 
j J da dvds cos\(a—x—ts)v-\-\i7r\v i s i l f{a), 
which is a deduction from Fourier’s theorem. 
Changing the notation (and in particular writing s and for his n and i) the 
method is here applied to the determination of the s-tuple integral 
'=^dx... 
n 
( X 2 ^ 2 \ 
(where <p is an arbitrary function) over the ellipsoid 
157. The process is as follows : we have 
<p(/2- + p) ! 
•••+ A2 — 1- 
{ (a- X f...+ (c-z)* + e 2 \* s+l1 (is + q) 
T 
du dv dTv is+q t is+q ~ l 
cos^u-^...~-r((a-x)\..+( c -z) 2 +6?)v)+i(is+q)7rj(pu; 
5 E 
MDCCCLXXV. 
