2-20- Of the SOLIDS 



by trial, that, when the function ~ L (r 3 -f Vi -f- r 6 ) is a 



maximum, r is nearly = _. Therefore, becaufe a ■=. — = li, 



5 r a 3 6 



6 2 c 

 r is nearlv to a as - to _ £, or as 216 to 12.5 ; and this of con- 



5 3 

 fequence, is, nearly, the ratio of the radius of the bafe, to the 

 altitude of the half-cylinder, when its attraction, eftimated ac- 

 cording to the hypothefis of the problem, is. the greater!: gof- 

 fible. 



XVII. 



To determine the oblate fpheroid of a given folidity which 

 fhall attract a particle at its pole with the greateft force. 



Let there be an oblate fpheroid generated by the revolution 

 of the ellipfis ADBE (PI. 7. Fig. 10.), about the conjugate 

 axis AB, and let F be the focus ; then if AF be drawn, and 

 the arch GG defcribed from the centre A, the force with 

 which the fpheroid draws a particle at A, in the direction AC, 



is 4*-AC.CD* ( , CF _ CG *) t (Maclaurin's Fluxions, §650). 

 Let this force := F, AC =z a, CD •= b, the angle CAF = <p ; 

 then CY ~ a tan <p, and F = ^f ^ (tan <p — <p) a =. 



4 n b tan <p — (p. 

 a tan <p 3 



Now if m J be the folidity of the fpheroid, fince that folidity 

 is two-thirds of the cylinder, having CD for the radius of its 



bafe, 



* The multiplier a t, omitted by Maclaurin, is reftorecT as above. 



