86 MR. C. J. BARGREAVE ON THE CALCULATION OF ATTRACTIONS, 
The first term is 
— 2 % % \a 2 — — — — r 2 ), ( e being the eccentricity). 
The third term is 
f KteJ'J'J'-i (1 - f* 2 ) (t - ^ydr’d^ 1 
= - ? K 2./- 1 (t - f* 2 ) (I - I*" 2 ) r2 l °s (-^r- + §r)<V- 
K 2 /l 9 \ 9 ( 2 ca 2 A 1 v'(fl 2 — C 2 ) 2 c 2 4\ 
= -x fT 3~ — f*V r2 VT (fla _ c i)l tan ~‘ c T^?--9F 
= — 3 *e (t - ^) r! (t — ■? + ^ (1 y C - Sin-’e), for K 2 = 9. 
4 7T p 7 
The value of V for the sphere of radius r, calculated by the usual method, is — ~ — ; 
consequently, for the whole ellipsoid, the value of V is 
4nf>r 2 ( \/(l— e 2 ) . \ ( 1 / 1 1, '/(l— e 2 ) . \ 
—g b 2 ‘rg{a 2 sm~ 1 e - r 2 ) — 3 tt§ ^y - p 2 ) r 2 ^y — y H y — sm -1 ej. 
Differentiate to f, by means of the equations r 2 = f 2 + g 2 + h 2 and g = — , and we 
have 
dY . I \/(l-e 2 ) . , \ 
-jj, = attraction in4 , = +2‘Tf/^l — yd y — sin - 1 e J ; 
dV 
which are the 
common ex- 
/I \/(l — e 2 ) 
so — -j-^ — attraction in y — + 2 n § g yl — y + y — ■ sin - 1 e ) ; )■ pressions 
dg 
dY 
dY . , , / , 1 
so — = attraction m y — + 4tt g h y+y 
^/(l-e 2 
sin - 1 e) ; 
otherwise 
found ■jf. 
Also 
dY 
d r 
= attraction to centre = +4 jy+-|-’r(y — ^XlT - ^ 2- ^ 
■v/(l-e 2 )„ : 
3 sm - 1 e 
)}■ 
10. By a similar process, I have deduced the attraction of an oblate spheroid, on 
a point within it ; the density varying inversely as the distance from the centre. The 
corresponding expressions are 
dY , n f 3 
~ 7/7 — + 27r £ — 
2 I - e 
d f 
dY 
dg 
dY 
- 7/77= + 27r £ 
r 4 a ( 
£ 3 
\ l-ey- - 
2 e 3 3 e 
g/ f 2_ _ J_ _ /( 
1 - e 2 ) | e 2 3 V 
ZLiiLfJ-__L_ A 1 " 6 !)! , Anf) ipo-Ltfl. 
/• 4 0 (1 - c 2 ) j e 2 3 V 2e a ‘3 e / s 1 — e J ' 
* See Pratt. Mec. Phil., § 172. 
t Ibid. § 158. 
