Prof. Challis on the Theory of the Moon's Motion. 375 



e 2 m 2 

 2 + 1" 



But by the above value of e 2 , _ = 1 — — ■ + ^-nearly. Hence 



it follows that 



. 3m 2 

 c=l+— . 

 4 





Here it is to be remarked, that we have arrived at a value of 

 c which is not true to the second order of small quantities, the 



true value, as is well known, being 1 — . The explanation 



of the above result is, that the approximation has not been car- 

 ried far enough to determine the value of c to the second order 

 of small quantities. The first term in the development of the 

 radius-vector which contains c is e cos (cnt + e— •or) ; and if, after 

 putting 1 + « for c, this term be expanded according to powers 

 of «, the second term of the expansion contains the factor ecu, 

 which if a be of the second order, is of the third order. Conse- 

 quently, as our approximation only extends to quantities of the 

 second order in the development of r, it does not embrace the 

 value of c to quantities of the second order ; it only proves that 

 that quantity contains no term of the first order. Hence, as far 

 as this approximation shows, c=l. 



In the paper already referred to (p. 528), I have shown how, 

 in the second approximation, to take account of terms which rise 

 in value one order by integration, and have obtained the follow- 

 ing complete values of r and to the second approximation : 



r e e 15?w£ 



-=1— ecosp + - — — cos2//— m 2 cos2g ^— cos (2q—p) 



. . a ■ . 5e 2 . „ llro 2 . n 15me . ,_ 

 cp = q + 2esmp + — snTp-l ^— sin 2^4- —t- sm{2q—])). 



These are the values to be employed in proceeding to the third 

 approximation. 



Before entering upon the next approximation, I propose to 

 indicate another method of obtaining the above values of r and 

 <f>, which will be found to be convenient in the higher approxi- 

 mations, and adapted to the consideration of the terms that rise 

 in order of value by integration. In this method it will first be 

 necessary to inquire in what way terms of that kind are intro- 

 duced. It is plain that they must be terms the argument of 

 whose circular function is a multiple of q— p, and that, as shown 

 by equations (A) and (B), such terms rise in value by integration 

 when they occur either in the expression for ? s sin 2<£, or in that 



7/1 



for -jr. It may be readily shown that terms of the form A cos2/> 

 in the value of r, and of the form A' sin 2p in the value of 0, 



