208 ON THE SOLUTION OF EQUATIONS 



modification, the new value of U Q will be unity, and therefore 

 we have for the initial and final values of u x , 



M = l, w a =0 ...................... (25). 



The necessity of this modification arises from this, that otherwise 

 the relation expressed by (24) would not be in all cases true. 

 For when x = l, we should have u^ qu^ whereas the true value 

 is of course 



From (24) we have (introducing the relation p + q = 1), 



(26), 

 a and /3 being arbitrary constants : and thence, by (25), we get 



and consequently u x - * ........................ (27). 



This expression is therefore, except for x = 0, equivalent to 

 (23), into which however the relation already mentioned, 

 viz. that p + q 1 has not as yet been introduced. 

 When p and q? are equal, (27) becomes 



while (23) similarly becomes 



. r . rx 



sin - TT sm TT 



1 5* a-l a a 



U *~a Zl ~ ' 



1 COS - 7T 



a 



or u x = - Sj ' 1 cot - ^ sin TT (29). 



a a 2i a 



Comparing (28) and (29), we have the following theorem: 

 writing x for a x* 



x = 2- l cot- sin -TT (30), 



a i a 



the upper sign to be taken when r is odd. 



