\ 



r 



114j 



BowditcVs Remarks on Doctor Stewards Formula. 



plane of the ecliptic^ 



tricifcies e. e\ of the 



may neglect the third powers of the i 

 its of the moon and earth, and the 



depending on 7, the tangent of the inclination of the lunar orbit 



the eclii 



Then bV La Place's notation, v represents the 



angular motion of the moon round the earth in the time t ; m the 

 ratio of the periodic time of the revolution of the moon about 

 the earth to that of the earth about the sun ; (1 — c)v the motion 

 of the moon's apsides, and by page SI 3, vol. iii. Mecanique Ce- 



leste c 



Vl 



^ 



J) ; J) being the coefficient of — e • cos (c v 



'Z*r 



)ia 



the equation (U) page S09 of the same volume multiplied by .-7—, 

 or by tty because we neglect terms of the order e^ ; so that if we 



a 



put, as inpageSS7; of the same work m . — = w^, this value of ^ 



will become, by neglecting terms of the fifth order in m 



9 



V 



\m\{% 



the values of Aj^^ Ai^^^ being determined by the following equa- 



It 215 of the same volume, in which we have 



tions given m page 215 



neglected e-, e'^, and some of the terras producin 



?«* in jp 







ii 



4(1 



mfl,K^'^-^^mK{i + 



1 



1 



m 



-V! 



(2) 



0= { l-(2_3m-c) ' J .A/o+Sm'. | > ^(i+M+ >Z^^___ 



2(l+m) 



(0 



4 



?/t— c 



^A,'>+A,^^' 



v> 



If we neglect the terms of A/^ producing m^ in j?, the equation 



(2) will give 



A 



m^. 



The first term of ^i in the equation (1) is the most important. 



nd. if 



only this term 



shall have p 



w', and 



c 



V\ 



(B) 



f 



shall 



1 



J ^^2 



4 



w 



} 



■ly. Substituting this in the equation 



for Ai^^^ the following 



'} 



glecting terms of 



the order m'; which produce w' in 7?, 



