62 
Mr. Ivory on the Attractions of an 
case requires, we shall have 
K'dR' 
dx 
dy 
d R' 
V&i — y'—z? COS 6 
R a sin, 6' cos. 0' . <f0' R' cos. 0'. rf0' 
V' R' 1 sin. a a'_ x 1 
cos. TO 1 
dz — R 7 sin. 9' cos. vr . dv r: 
consequently (the density being denoted by unit) dM — dx . 
dy.dz = R 7 \ dR ' . d9' sin. 9' . dvr. Farther, let a — r cos. 9, b = r 
sin. 9 cos. vr, c = r sin. 9 sin. w; then, by substitution, 
# v __ yy y* R a . <7R' . rfQ' sin. 9' . duf 
A J J V r a — 2 rR' . (cos. 0 cos. 0' -J- sin. 0 sin. 0' cos. (to'— to)) 4 - R' a 
and if we put cos. 9 — cos. (3 7 = p; then 
y __ rrr R a . d R 1 . dy . dJ 
jJJ v?~ 
Vr a _ zrR' . y -f R 2 
(4)- 
7 = ^ 1 / -|- \/ 1 — . v 7 1 — !«/* . COS. {vr' — vr) 
7. When the attracted point is without the surface, the ex- 
pression for V, in order to embrace the whole mass of the 
spheroid, must be integrated from R 7 — 0 , to R 7 = R, R de- 
noting what R 7 becomes at the surface ; from f — — 1 to 
and from vr' = 0 to vr' = 2v, 2v being the circumfe- 
rence when the radius is unit. In this case V must be reduced 
into a series containing the descending powers of r, which we 
may thus represent, viz. 
V = 
b( o) , b(‘) , b( 2 ) 
+ 
B(h 
J + 1 
&C. 
and if we expand the radical in the last expression of V into 
a similar series, and use to denote the same thing as for- 
merly in No. 4, we shall get, by equating the corresponding 
terms, 
t * Mec. Cel. Liv. je. No. 8. 
