116 BowditcK^s JRemarfcs on Doctor Stewards Formula, 



be of 



order m\ but 



creased by a divisor of th 



order m, with which it was affected, when its value was calcu- 

 lated by meaus of the equation (3) ; this made it too important t» 

 be neglected, as had been supposed at first might be done. 



Doctor Stewart's theorem, expressed as above in an analytical 

 form makes the motion of the apsides in one anomalistic revolution 



equal to 360 



F 



360% or by development according ta 



the powers of 



F 



/ 



1 



F+|-^ + ^=-}-360 



(8) 



and if we use the 



F 



y^ 



J 



w', as it is given by Doctor Stew 



Prop. 19 of his fourth Tract,* it becom 



3, 



4 



n^^ IS-f- V .m\ ? . 860 



(9) 

 ivhicb differs very much from tliat computed in formula (7). 



For 



we find, that if m 



by comparing the t\to expressions (7), (g), 



very small, Doctor Stewart's theorem makes the motion marU 



double 



valu 



If w is nearly equal to 0.08 ., ... the 



numerical values become equal ; but if m exceeds that quantity, 



his theorem makes 



motion 



small. Th 



f 



would he altered a little by the introduction of the terms we have 

 neglected. This limit is very nearly equal to the value of m cor 

 responding to the moon^s orbit, and from this accidental circum 



* This is given as an approximation towards the true value, and it agrees 

 nearly witb Mr. Landen's estimate in page 9 of his Animadversions, The terms 

 of a higher order in m, which were neglected by Doctor Stewart, would net 

 affect the first term of formula (9), it was therefore unnecessary to correct 



/ 



term, as well as it could he by the comparison of the whole series, 



first 



X 



I 



^ 



.*■ 



\ 



