184 ON THE MOTION OF THE MOON'S NODE [24 



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 term 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. Meantime I now make known the result which I have obtained 

 for the motion of the Moon's node on the suppositions stated in the title 

 of this paper. 



If nt and n't represent the mean longitudes of the Moon and the Sun 

 at time t, omitting, for the sake of brevity in writing, the constants which 

 always accompany nt and n't, and if 6 and r represent the Moon's longitude 

 and radius vector, I find that, in the case of no eccentricities and inclination, 



n' 



if m = = 0'0748013, which is the value used by Plana, 



0-01021,13629,5 sin 2(n-n')t 

 + 0-00004,23732,7 sin (n-n'}t 

 + 0-00000, 02375, 7 sin 6(n-n')t 

 + 0-00000,00015,1 sin S(n-n')t 

 + 0-00000,00000,1 sin W(n-n')t; 



-= 1-00090,73880,5 

 r 



+ 0-00718, 64751, 6 cos 2(n-n')t 

 + 0-00004,58428, 9 cos 4(n-n')t 

 + 0-00000,03268, 6 cos 6(n-n')t 

 + 0-00000,00024,3 cos 8 (n - n') t 

 - 0-00000,00000,3 cos 10 (n - n') t ; 



supposing that is expressed in the circular measure, and that the unit 

 of distance is the mean distance in an undisturbed orbit which would be 

 described by the Moon about the Earth in the same periodic time. In 



