2. 



DEVELOPMENT OF A CERTAIN INFINITE DETERMINANT ARISING IN 

 RELATION TO THE MOTION OF THE NODE OF THE MOON'S ORBIT. 



[THE aim of the following pages will be made clear by an extract 

 from a paper of Adams "On the Motion of the Moon's Node" (Hon. Not. 

 xxxvni. Nov. 1877 ; Works, Vol. i., p. 181). This paper was evoked by Dr 

 G. W. Hill's now famous work on the Lunar Perigee. It appears that one 

 part of the process invented by Dr Hill for evaluating the motion of the 

 perigee had already been found by Adams to yield the motion of the node 

 to a high order with rapid approximation. After describing, in the paper 

 in question, his views of the most advantageous method of treating the 

 Lunar Theory, and mentioning his early determination of the Variation 

 terms, he continues : " In the next place I proceeded to consider the 

 inequalities of latitude, or rather the disturbed value of the Moon's co- 

 ordinate perpendicular to the Ecliptic, omitting the eccentricities as before, 

 and taking account only of the first power of y. 



" In this case the differential equation for finding z presents itself 

 naturally in the form to which Mr Hill reduces, with so much skill, the 

 equations depending on the first power of the eccentricity of the Moon's orbit. 



" In solving this equation I fell upon the same infinite determinant 

 as that considered by Mr Hill, and I developed it in a similar manner in 

 a series of powers and products of small quantities, the coefficient of each 

 such term being given in a finite form. 



" The terms of the fourth order in the determinant were thus obtained 

 by me on the 26th December, 1868. I then laid aside the further in- 

 vestigation of this subject for a considerable time, but resumed it in 1874 



