44] 
285 
44. 
ON A MULTIPLE INTEGRAL CONNECTED WITH THE 
THEORY OF ATTRACTIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 219—228.] 
Mr Boole [in the Memoir “ On a Certain Multiple definite Integral ” Irish Acad. 
Trans, vol. xxi. (1848), pp. 40—150] has given for the integral with n variables 
f <f> (§- 2 + | 2 + -^jdxdy... 
v= I M y L nv 
J [(a — x) 2 + (b — y) 2 ... + u 2 ~$ n + q ^ ' ’ 
nn2 /o#2 
limits yi+|i + -<l. 
the following formula, or one which may readily be reduced to that form 1 , 
y fg ... 7r? n f 00 ds 
(2), 
r (in + q)J v J{(s+f 2 )(s + g 2 )...} 
where 
«-Vi 
+ t (1 — <r)} dt 
■••(3); 
in which 
a 2 b 2 
f 2 + s ' g 2 + s ' 
u 2 
• • H— 
s 
G) 
and tj is determined by 
a 2 b 2 
f 2 + v 1 g 2 + V ’ 
u 2 
■. H— ■ 
V 
1 See note at the end of this paper.