548 

 which gives, by summing the two geometric series, 



We get from (6) 



and the first member of this equation being symmetrical with respect 

 to m and n, we get, by interchanging m and n and equating the results, 



or 



which is therefore constant. Denoting this constant for convenience 

 by 2 cos a, we have 



(4/z) 2 =l 2cosa.0(m) + (^) 2 ..... (8) 



Squaring (7), and eliminating the function ip by means of (8), we 

 find 



{l-2cosa . <Kw) + < 2 } . (9) 



From the nature of the problem, m and n are positive integers, and 

 it is only in that case that the functions ^>, ^/, as hitherto denned, 

 have any meaning. We may, however, contemplate functions 0, $ 

 of a continuously changing variable, which are defined by the equa- 

 tions (6) and (7) ; and it is evident that if we can find such functions, 

 they will in the particular case of a positive integral value of the 

 variable be the functions which we are seeking. 



In order that equations (6), (7) may hold good for a value zero 

 of one of the variables, suppose w, we must have 0(0)=0, ^ (0)= 1 . 

 The former of these equations reduces (9) for nQ to an identical 

 equation. Differentiating (9) with respect to n, and after differen- 

 tiation putting w=0, we find 



(w) { 1 2 cos a . 0(m) + (<$>mf j- + cos a . <p'(m) $(m ^(m) 

 2 cos a . 



