144 Mr. Booue on a certain Multiple Definite Integral. 
n Ld d ‘= n p 
t+ hifi wage ds (=)"* ¢(0) rs 
r 5 = 5 1 3 ’ = 
@ | |e 
and o being the symmetrical functions of s given in (11), and f(o) being re- 
placed by 0 when o > 1. 
The reader who is acquainted with the researches of M. Dirichlet and Mr. 
v =(-) 
Ellis, will, in various parts of the preceding investigation, be reminded of them, 
and I am anxious to express in the fullest manner my obligations to them. But 
I think it will be evident, that while the methods of those writers apply to classes 
of questions which are mutually distinct, neither of them is, in its actual state, 
applicable to the question of this paper. 
PARTICULAR DEDUCTIONS. 
Ist. Let n = 3, 7 = 4, and let us write xyz for «7,2, and abe for a,a,a,, 
we have 
dudydz f (FZ m + o — taal 3 
= Wiese i 
subject to the inequality, 
xr Y 2 as f 
het fab pee 13 ue 
and its value given by (12) is 
ds(=) J (2) 
= —hAh.h; ee eee 15 
1 EN TER) (ERE) (EADIE oa 
where ae bP & 
x mae nrr ss sae sth ey 
Now the attraction of an ellipsoid of variable density, on an external point, the 
law of force being that of nature, is expressed by the integral, 
dadydz ( (a—a)f (FZ +h are a) dy 
Mie 2} + (b= ae zy a 
= hhh S da \de iF F (0) 
fo 2)(s-Fhe) (6A) 
