90 Mr. Ivory on the figure requisite to maintain the equilibrium 
V (r) = R’x { P° + P<‘> . ±r + P W - £ + &c - } > 
the coefficients P°, P' l> , P' 2J , &c. being all functions of p, 
Vi — Cos . T3-, V i — [x* . Sin. csr. When the attracted point 
coincides with the origin of the co-ordinates, the value of 
is equal to ; and when the same point is in the sur- 
face, then ~ = 1 , and the value of is equal to 
p (°) + p(0 + p ( 2 ) + &c 
Finally, let the attracted point be without the surface ; then 
the quantities common to all the bodies are these, viz. 
— , — : hence 
r r 
v O’) „r fR a_ b CT 
R a ~ 1 r ’ r ’ r ’ r j ' 
Consequently, 
V(r)=R>xF. i\; 
V (r)=R s xF. [x, ^ l — [x 9 Cos. nr, Vi — ^ . Sin. % a- j • 
In this case, ~~ decreases as r increases, and finally 
vanishes when r is infinite. The expansion must therefore 
have this form, viz. 
V (r) = R* X {Q (,) . y + Q <j) . £ + Q (3) T + &c. } , 
Q (l) , Q (2) , Q (3) , &c. being functions of V i — p*. Cos. 
V i — [x 2 . Sin. When the attracted point is in the surface, 
~ = i , and the value of — is equal to 
Q (,) + Q (2) + 0 (3) + &c- 
The preceding reasoning is quite general, and will apply 
