114 NUMERICAL DEVELOPMENTS IN THE LUNAR THEORY. [3 



We next find the parallactic inequalities. We have the equations 



#0 (ttde 

 dip + dtdt 



+ n'" - sin 2o 4- Xi' 2 - 3 sin w + - sin 3co = 0, 



[_ z J o |_o o 



. E-Xa 

 where X 



restoring a which was taken as unit, and is defined by 



We have found values of I, 9 which reduce to zero the sum of all the 

 terms in each equation excluding those multiplied by X ; let the result of 

 substituting these values of I, 6 in those terms be X, Y ; then our equations 

 for correcting I and will be of the form we have just discussed, and 

 we might proceed in a like manner. But we can see that the approxi- 

 mations would be comparatively tedious ; for, taking the equations 



at" at 



d'S^ai . dSJ 



suppose 



P l = tpj cosj (n n') t, 



Q r = tqj sin j (n n'} t, 



where j takes all positive odd integral values ; then let 



8^ = Zcij cosj (n n') t, 



Sjco = Zbj sinj (n n') t, 

 and we have 



= PJ j-(n n'Y a>j 2jn (n n') b } 3ca,j, 

 = qj j' (n n')' 2 fy 2jn (n n') a, , 



Multiply the second equation by 2 . K and subtract from the first ; 



j (n n') 



