25] RECIPROCAL OF THE MOON'S RADIUS VECTOR. 195 



Now it is known, a priori, that the values of r and u, as well as that of 



- , may be developed in an infinite series involving cosines of angles in the 



form 2;f/< +yy+ 2^, 



where i, j, f, and k denote any positive integers whatever, including zero, 

 and that the value of v may be developed in a similar series involving 

 sines of the same angles. 



Also we know that the coefficient of the term with the above argument 

 occurring in any of these series contains eV^'y 24 as a factor, the remaining 

 factor being a function of m, e 2 , e'" and y". 



Similarly we know that the value of z may be developed in an infinite 

 series involving sines of angles of the form 



and that the coefficient of the term with this argument contains e'e tf y tk+l 

 as a factor, the remaining factor being a function of m, e 2 , e'* and y 2 as 

 in the former case. 



It is essential to observe that - , r, u, and v involve only even powers 

 of y, while z involves only odd powers of the same quantity. 



Having made these preliminary observations, we are now in a position 

 to apply our principle to the four cases already alluded to. 



CASE I. 



First, suppose that the values of x, y, z are those belonging to the 

 solution in which e, and y vanish, therefore all the arguments in the values 



of -, r, u, and v will be of the form 2i(j f ^ and z will vanish. 



Also let the values of x u y v z l belong to the solution in which e has 

 a finite value, but y is still = 0, while nt + e, and therefore also n, retains 

 the same value as before. 



Hence z l also vanishes, and therefore z z l Q. 



252 



