,, 4 DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



X for which there exist solutions of (2) satisfying 



The values of /*, say A,, A, ..... A., > T7 11P tW if if, M 



B' are known to be the roots of the determinant of *<*,4 Further if ^ (,s), 

 , .... *.(*), .- respectively are these solutions, chosen m such a way that 



g = l ( n=l > 2 ' ')' 



it is known that they are the complete system of normal functions of ir(M> 

 Recalling that X 1? X. .., A., ... are all positive, it follows from the theorem quoted 

 above* that , . 



= 2 4^1M) ......... . (4) 



4. (Consider now the effect of replacing q by q + \ in (2), where X is a negative 

 number, t The values of /i for which there exist solutions of 



satisfying B' are clearly X,-X, X,-X, ..., X.-X, ..., and the solutio ns corresponding to 

 these are fc(). *,(), .... *.(*)> .... respectively. It follows from (4) that the 

 GREEN'S function of 



oV 

 is 



n X 



i.e., K A (s, t), the solving function corresponding to K(S,I). But, by KNESER'S 

 theorem, the GREEN'S function of (5) for the pair of boundary conditions B' is the 

 function denned by 



M)* k () (*=:*) 



where B A (.s) satisfies the equation (5) and the boundary condition 



du , , 



h'.u = at s = 0, 



as 



and <I> A (s) satisfies this equation and the boundary condition 



-= h H'w = at s = TT, 

 as 



* II., 1. 



t For our immediate purpose this restriction may be replaced by the wider one that X is not equal to 



one of the singular values A,, X 2 A n , .... As the condition that X should be negative is forced upon us 



in the following paragraph, and we shall not need to consider other values, we shall continue to suppose 

 that A is negative (ride II., 3). 



