88 DEVELOPMENT OF A CERTAIN INFINITE DETERMINANT ARISING IN [2 

 Hence the above equation becomes 



which gives 



27r<r 



cos2y7r = cos2K7r + r - lN /0 prSm2K7r. 

 /c (2/c 1) (2/c+ 1) 



(26 Dec. 68.) 



[On comparing the above with the paper "On the Motion of the 

 Moon's Node, &c." Man. Not. Nov. 1877, it will be seen that in addition 

 to the convention there adopted as to the unit of distance, the unit of 

 time is so chosen that n=l; also 2/f = </, 4rt = </,, 2y = g. In the subsequent 

 work this change of notation will be introduced. 



The subject was resumed in 1874, when we find the following entries 

 in the Diary : 



Feb. 4. Both yesterday and this morning while in bed thought over 

 the mode of treating the linear differential equations which occur in my 

 way of investigating the lunar inequalities. Think I see my way. In the 

 morning worked at one part of the subject. 



Feb. 7. Worked nearly all the morning at formation of terms of 4th 

 order of my determinant. In evening finished calculation of terms of 4th 

 order. 



Feb. 10. Thought over method of treating mean motion of apse 

 similarly to that of node by means of a differential equation of 2nd order. 

 Began operations by transforming fundamental equations into one with 

 (f) = n't for independent variable. 



Feb. 11. Went on with my investigation so as to form the differential 

 equation of 3rd order in Av and the theory of its reduction to the 2nd 

 order. 



Feb. 23. Thought of a simpler mode of treating the differential equation 

 for A0. In evening worked out reduction of equation to 2nd order in an- 

 other way. 



With respect to the method adopted for developing the determinant 

 to higher orders, we notice that we may either proceed entirely along 

 the diagonal, which gives unity, or forming a minor determinant with 

 any number of consecutive constituents of the diagonal of the infinite 



