25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 191 



These investigations of Plana and Ponte"coulant, however, while they 

 shew that the coefficients of the above mentioned powers of m vanish by 

 the mutual destruction of the parts of which each of the coefficients is 

 composed, supply no reason why this mutual destruction should take place, 

 and throw no light whatever on the values of the succeeding coefficients in 

 the series. 



Thinking it probable that these cases in which the coefficients had been 

 found to vanish were merely particular cases of some more general property, 

 I was led to consider the subject from a new point of view, and on 

 February 22, 1859, I succeeded in proving, not only that the coefficients B 

 and C vanish identically, but that the same thing holds good of the more 

 general coefficients B and C, so that the coefficients of 



e 2 , c-e''\ e'e", &c. 

 /, ye'-, ye", &c. 



in the constant term of -- are all identically equal to zero. 



Further reflection on the subject led me, several years later, to a simpler 

 and more elegant proof of the property above mentioned. 



This new proof was found on February 27, 1868, and I now venture to 

 lay it before the Society. The resulting theorem is remarkable for a degree 

 of simplicity and generality of which the lunar theory affords very few 

 examples. 



There are also two remarkable relations between the coefficients of e\ 

 e'y, and y* in the constant term of - , which we before denoted by E, F, 

 and G. These relations may be thus stated : 



If the terms of the quantity c or -~ which involve e~ and y 2 be 

 denoted by 



and similarly if the terms of g or 4-. which involve e 2 and y be denoted by 



where H, K, M, and N are functions of m and e' 2 , then we shall have 



E_H f^_M 



F ~ K and G~N" 



