[ 223 ] 



where a is an arc thecofine of which is — — -ZI'^_rad.being-i. 



e ex 



Hence for every dift. S P we have the angle ASP, and confe- 



quently the orbit is determined. When the body comes to the 



o o 



lower apfid, a—i^o ,-. the angle between the aplides -• i8o 



\/^-^ ^ -'x I- ''' + ^'''' , , &c. 



I — j\.c-rce^ 4.1 — ^c-\.ez c 4.1 — /^c + e^dj- 



o . 



The limit of this quantity is 180 JJ*^ t- (2 Cor. 45 Prop.) 



-¥ 



This propofition is applicable to the lunar orbit. The 

 limit of the refult is the fame as found by Sir Ifaac New- 

 ton. Some authors have conceived Newton's conclufion er- 

 roneous, and with the fan?e law of force have found the mo- 

 tion of the apfides twice as great. Walmfly, particularly, has 

 imagined, that the principles of the gth fedion give tha true 

 angle between the apfides only when the force varies accord- 

 ing to a iimple law of the diftance. In his trad, " De 

 " Inaequal. Lunas," he finds the motion of the apfides, by comput- 

 ing the time a body takes in acceding towards the centre a fpace 

 equal to twice the excentricity, when impelled by a force 

 which is the difference of the centrifugal and centripetal forces. 

 This time he compares with half the periodic time, and thence 



E 2 ^ deduces 



