23^^ J 



apfiJ,(:=: i8o nnd s, a;s, z a &c. ~oy the angle between the 

 apfides = , . 



'iSo_x I + " +'^' " J' e'- , &c. in which only the even 



r 1 ■ 2. "? 4. 



\ « + 2' - ^ i-' 



powers of the eccentricity can enter. It is evident from this ex- 



preflion that the motion of the apfides will be always affeded by 



the eccentricity of the orbit, unlefs either « ^ i = o or n — 2—0 or 



tha.t the force varies either in the inverfe duplicate ratio of the 



diflance or in the diredt fimple ratio. We know from other 



principles that in thefe tvpo laws the eccentricity does not affed 



the angle between the apfides, but it is a very remarkable cir- 



cumftance that this takes place for no other law. 



The above propofitions point out how the orbit may be 

 found whatever be the fundion of the diftance expreffing the 

 centripetal force. For z will be always found a fundion of e, 

 and therefore/ will always be a fundion of ^ and f, confequently 

 f!_ — I will be a fundion of x and e, or fubftituting for x, i-^y, 



1_ — I will be a fundion of y and e. And two roots of the 



pi 



equation 



