25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 199 



That is, 

 is divisible by (e e?)* ; or 



is divisible by (e 2 eff. 



Divide by e 2 e, 2 and then put e^ = e", 



TT 



therefore - 2^e 2 + 2F- e* = 0, 



E H 

 or ~ 



CASE IV. 



Lastly, let the values of x, y, z belong to the solution in which c 

 vanishes and y is finite, while x l , y,, z l belong to the general case in which 

 e and y l are both finite, the value of y being changed to y^ while nt + e, 

 and therefore also n, retains the same value as before. 



Then all the arguments which occur in the values of -, r, u, and v, 

 and which are of the form 



will occur unchanged in the values of -, >\, u 1} and v lt provided that 77, 



?*j 



and therefore also %- or g, remains unchanged, but the coefficients of the 



corresponding terms will differ by quantities which involve either e 2 or 

 y*~7i as a factor. 



Let the terms with these arguments be called terms of the first class. 



Also there will be additional terms in the values of , r 1} u 1} and v lt 



r, 



the arguments of which are of the form 



where j does not vanish. The coefficients of these terms will all involve e 

 as a factor. 



