90 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
tion between the surface and that spheroid on which the point lies, whose ellipticity is s v 
The ( n + 1 ) th term of V now becomes 
2 » S rf- i P » d t>' { fr dr ' + ^ fr d r ' } • 
a<p t a — <p lt & — y a 2 (p a — y (y, 2 — y) r 2 <p a 
- (a <p, a — <p u a - a 2 <p a - -£■ - -|) r 2 <p a) 
The first part of this gives 4 n 
a being semiaxis major of the stratum on which r lies. To determine the other part, 
it is necessary to compute it when n — 0 and n = 2, which gives 
4ir e {\JH d7ddd + T r 2 (^-r)X‘^- da 'j- 
To the sum of these we must add the value of V for the inner spheroid ; and for this 
purpose we have to obtain V for an external point. 
To the expression in (13.) we must add 
o ' 
PJl'i d f J (pP P, _ F r 1 r^'d r) , 
to be calculated when n = 1 and n = 3. This is 
r rf-~ 
^e{hfy d - dtdd + -^ f 0 "* d - didd 
} 
The whole value of V is 
^a--^a 2 <pa- y(^ 2 — y) s^<p a + y f Q F a' . a' 2 d a! 
4 7T g 
/** - 
+ \ f a F a! . a' 4, d 
1 5 r- Jo 
After writing a for a, and s x for s, add this to the other value of V, and apply the equa- 
tions 
F a' . a' d a' a! 2 s' cl <p a! — a 2 s <p a — a 2 s x (p a — <p a! d ( a 12 s'), 
F d . a! 2 da' — \ a ' Z d P a ' = s { <p d. — ~ <p a! d ( a ' 3 s'), 
and similar equations for the other integrals ; and we shall obtain 
fl? ( a-^fl-a^a+^a + 7 - f* <pa! d(a' 2 s') + <pa'd(a'h') J I 
V = 4 7Tg\ L M* 
“ y (^ 2 ~ y) {f^ 6!di + ^X^ a ' d ^ i ') } j 
