Prof. Challis on the Eccentricity of the Moon's Orbit. 137 



commences from a fixed ellipse. Since the article was written, 

 I have ascertained that no failure occurs on that account in the 

 approximations to the first and second orders of small quantities, 

 the results being so far the same, whether the approximation 

 commences from a fixed ellipse, or is conducted by the process 



which I have shown to be necessary if -^ contain the disturbing 



force as a factor. This statement may be verified as follows. 



Let the force be wholly central, and equal to fiu 2 ——, u being 



the reciprocal of the radius-vector. Also let fj! be a small quan- 

 tity whose powers above the first are neglected, and the orbit be 

 nearly circular. By a known equation, we have 

 d% /u, fj/ 



Omitting the small term, let the result of the integration be 



c(l— e 2 )w = l + ecos#. 



Substituting this value of u in the small term, expanding, and 

 omitting powers of e above the first, 



d 2 u fx fJa 3 Sa'a 3 e n 



-jM +«- JJ + —- = .r cos 0; 



h* 



or, putting for shortness' sake w for u— ~ + ~-, 



hi /,2 



d*w , 3u,'a 3 e n 



^. + «,= -_ cos 0. 



Now if this equation be integrated by the ordinary method 

 applicable to exact equations, the integral will contain a term 

 which may increase indefinitely with the time, just as in the 

 lunar theory. But no such result follows if the two integrations 

 be performed in succession, and regard be had to the rules of 

 approximation. After multiplying the equation by 2dw, and 

 putting for dw, on the right-hand side, its approximate value 



sin 6 d6, we get by integration, 



dw\ 2 3ftW . 



Hence, since approximately, 



dd ^C^uJ= -dw and C sin 2 0=C-w; 2 , 

 it follows that 



M= _ dw _8^W 



v^C-m, 2 2A 2 C ao ' 



