88 
MR. C. J. HARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
When n — 2, the term is 
f * 1 4* 2 - T )/- 1 ( ** - f ) (¥ + IT + 2 *£ + 2 • 3 + . . .) if l 
= Y r e r 2 {y? - j) 2, [to - J (f/ 2 - y) X“ f/ " ; ’ 
- 1 -r- (,•- 1K[- - .]/: 
= 5 t? r 0*“ — 3") e2 2 1 ~ 1 ] m \ ) = — y ** § r2 (^ 2 — y) e 2 <p a. 
To calculate V for the sphere whose radius is r. 
Let the sphere be referred to polar coordinates, the centre 
4 , being - the pole. 
Let O C = r, O P = r l5 and POC = 9 ; then P C = \/ (r 2 + r x 2 
— 2rr 1 cos0). 
Mass of the element at P = r x 2 <p r x d r x sin 6 d 6 d co, and 
V = 
r f* f* 2 n P r \ r \d r l sin 0 d 0 d co 
J o J o J o v' ( r 2 + r-? — 2rr x cos 0) 
*) 
/‘ r f >7r r x cf) r } sin 0 d r Y d 0 
^ o «./ o V (r 2 -f- r 2 — 2 r r x cos 
= rA ^p (( r + r i) ~ (r- rO) dr x = r/ 2 <p </ r x 
= ( r2 <P,r—2r <p u r + 2 <p m r — K), 
K being the value of <p x r x — 2 r x <p u r x -j- 2 <p ll; r x , when r x — 0. 
The whole value of V then is 
4 if fa fra - </>„a - ^-e 2 a°-<pa - (<p„r - yip,,, r + ^ r 2 (ft 2 - ^-) e 2 <p a j . 
4*-f{^ + y(f<. 2 - 3 )re'-<pa] 
And the attraction toward the centre 
dV 
dr 
where 
/ r 
r 2 dpr d r. 
13. To find the attraction of the same spheroid on a particle without it. 
The series (Art. 3.) is 
/»R /’l / /' ,3 \ 
27r Uo f- i < / )r '( P o 7 ’+ P i^*+ • •) dr dpJ. 
Now 
yv. r w + 1 . c? r f = r' n + 1 r' — (w -f- 1 ) r' ra <p tl r' n (n -\- \) r’ n 1 <p w r' — &c. (A) 
and the general (wth) term of V is 
A-, 0 
y* ,w "f ^ cp^ jd 
— (n - f I ) <p {l r 1 + « (re + 1) r'" 1 (p tjl r 1 — &c.^ d p! 
