the Semi-axis Major of the Mooris Orbit* 345 



pairs, of which the following is one, and may be taken as a type of 

 the rest : 



DBA f I f ~\ 



4 n \i" — i-\- j" m —j m-hl" —I i' + i+ J m +jm-t-l , + lj 



sin [Pn t -\-j' m nt + l' nt—i n nt + int —j n m n t+jmnt— I" nt + lnt). 



In order that this argument may be of the kind under considera- 

 tion, we must have 



i' — i" + i = 

 f m —j ,! m +j m = 0, 



and the term becomes in that case 



DBA f t}_ 



4?j \i" — i +j"m —jm + I" — I 



- •/ , TH T-' rirr>~V sin W -I" + l)nt 



v + i +jm +jm+l' + lj v ' 



if i is not equal to zero, D is of the order m% and the coefficient of 

 sin {V — I" + I) n t, after a fresh integration, remains of the order 

 m\ If i = 



i' = i» 

 the coefficient becomes 



i'DBA f f m + jm + V + I —fm + jm — l n + I "\ 

 4>rc L (* v/ +j"m —jm + /" — I) (i 1 +'/ m +jm + l'+ l)J 



i'DBA{jm + l<-l»+2l} 

 "4w {*" + /' 7?Z — J w + /" — /) (*' +/ ?» +i w + I' — /)} ' 



but unless^' = 0, Z) is multiplied by #2, and in either case, in con- 

 sequence of the reductions which the numerator undergoes, the co- 

 efficient after a fresh integration remains of the order m A . 



It remains now only to consider the terms of which the following 

 is the type: 



/ a*R \ &R /v _ /dR\ 



Vd7dj)mj fe"Hg>> 



but as in le are here only included terms of the order in 2 , v 



