96 



a • — 3? «=:— 2(fj, aor a =2e s, 4w 

 T^ 4 



« — I2e a =z—2cs, a or a = — 2e t, 5« 



TTT 5 

 tsfc. ^c, Ufc. 



Hence azzm+a+a+kc.z=m—2es, m+2£ j, 2«— 2£ /, ^m+kc. 

 a 123 



C)^ E. D. 



Obfervation. It is well known that 

 ^T+e: s/i—e : : t, \c : t, U, -aye being the true and excentric ano- 



maly of a Planet. 



or/, ia=t, jcy /i+g 



now if for y^T+e be fubftituted i+e' and the flux, of the equat. taken. 



axi^''=—2e's, a which equat. is precifely of the fame form as that 

 from which the feries in the above theorem was derived and therefore in 

 the fame manner we have 

 a=c—ze' s, c+2e' s, 2f— 2£' s, 3C+2£ s, 4c— &c. 



T 2 3 4 



and,becaufe ^i+e=i+^, e'= e This is the fame conclu- 



I — e I — e' i+-v/i— «* 

 Con as Laplace has deduced* by a very ingenious ufe of impoffible quan- 



titles. 

 From the above theorem it hnmediately follows that the anomaly 



computed by Seth Ward's method. 



=»j— 2«, m+2e s, 2m— -2e s, ^m+2^s, 4»2— «c. 



"2" 3 4 



comparing this with the true anomaly, ^__ 



* Mecaniqw cekfte, tome i. p. 17°' 



