190 ON THE CONSTANT TERM IN THE [25 



a = (-^\ , the mean distance in the purely elliptic orbit which the Moon 



\n I 



if undisturbed would describe about the Earth in its actual periodic time. 



To fix the ideas, we will suppose the quantities e and y to be defined 

 as in Delaunay's Theory of the Moon. 



If r denote the Moon's radius vector, and if we omit terms depending 

 on the Sun's parallax, then, as is well known, the value of may be 

 expanded in an infinite series involving cosines of angles of the form 



where i, j, f, k denote any positive integers, including zero, and the co- 

 efficient of the term with this argument contains eV^'y 2 * as a factor, the 

 remaining factor being a function of m, e~, e'' 2 , and y". 



In particular, there is a constant term in - , corresponding to the case 

 in which i, j, j', and k are all zero, and this term has the form 



A + Be 2 + Cy 2 + Ee 4 + 'ZFe'Y + Gy 4 + &c., 

 where A = A + A /" + A /" + &c. 



&c. &c. &c. 

 and A M A 1 &c., B , B l &c., (7 , C l &c. are all functions of m. 



Plana and, after him, Lubbock, Pontecoulant, and Delaunay have developed 



the functions of m which occur in the coefficients of the several terms of - 



r 



and of the other coordinates of the Moon, in series of ascending powers of 

 m, and have severally determined, by different methods, the numerical co- 

 efficients of the leading terms in these developments. 



With respect to the constant term in - , Plana shewed that the quan- 



tities denoted above by _B and C , viz. the coefficients of e 2 and y 2 in the 

 above constant, both vanish when account is taken of the terms involving 

 m" and m 3 . Pontdcoulant carried the development of the quantities B and 

 C, two orders higher, viz. to terms involving m 5 , and found that these terms 

 likewise vanish. 



