the Semi-axis Major of the Moon's Orbit. 34)5 



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

 the rest : 



DBA f y_ F ~\ 



4w \_i" i + j"mjm + l"l i' + i+j' m+jm + l' + lj 



sin (i'n t +f mnt+V nti" nt + in tj"mnt+j m n tl" nt + luf], 



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

 tion, we must have 



i' i" + i = 

 j' m j" m +j m = 0, 



and the term becomes in that case 

 DBAf 



\_i" i +j"m jm + I" I 



~ i , _L > - T-' - T7TT-/T sin ( l/ -V + l)nt 

 V + * +jm + jm + I 1 + IJ 



if i is not equal to zero, D is of the order ra 2 , and the coefficient of 

 sin (I 1 I" 4- /) n f, after a fresh integration, remains of the order 

 wz 4 . If i = 



i' = i" 

 the coefficient becomes 



i' D BAf j' m + jm + I' + I j"m + j m I" + I ~\ 



I' + l)J 



(i" +j" m j m + I" I) (i 1 +/ m 



_ i'DBA {jm + I'-l" + 21} _ < 

 = 4 n {i" +j" m jm + I" I) (i' +j' m +jm + I 1 I)}' 



but unless j 0, D is multiplied by m, 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 4 . 



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

 is the type: 



but as in 8 e are here only included terms of the order m", 



