86 DEVELOPMENT OP A CERTAIN INFINITE DETERMINANT ARISING IN [2 



and 1875, and on the 2nd of December in the latter year I carried the 

 approximation to the value of the determinant as far as terms of the 

 twelfth order, or to the same extent as that which has been attained by 

 Mr Hill 



" The equation which I had obtained by equating the above-mentioned 

 determinant to zero differed in form from Mr Hill's, and on making the 

 reductions required to make the two results immediately comparable, I found 

 that there was an agreement between them except in one of the twelfth 

 order. On examining my work I found that this arose from a simple 

 error of transcription in a portion of my work, and that when this had 

 been rectified my result was in entire accordance with Mr Hill's. 



"The calculations by which I have found the value of the determinant 

 are very different in detail from those required by Mr Hill's method, and 

 appear to be considerably more laborious. I have not yet had time to 

 copy out and arrange the details of the calculations from my old papers, 

 but I hope soon to do so, thinking that they may not be without interest 

 for the Society." 



This intention was not fulfilled ; the details referred to appear for the 

 first time in the following pages.] 



[With the date 26 Dec. 1868 we find the following.] 



If we 

 finding z is* 



-j- 2 + z (I + + ! cos kt) - 0. 



If now 



z = c a sin gt + c l sin (g + k)t + c., sin (g + 2k) t+ ... 



+ c_j sin (g k) t + c_ 2 sin (g 2k) t+ ... 



we obtain by equating to zero the coefficient of each sine in the result 

 of substituting in the above differential equation 



1 m! 

 If we call -^ + ^ = 1 + + ! cos kt, where k = 2 2m, the equation for 



= . . . [(gr - 2k)* - (1 + a,)] o_ - \ O0. t 



= -\<W-* +[(<7-) 2 -( 



* [See Lecture XIV. on Lunar Theory, p. 64.] 



