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



= b 







(iv) r = 



r ';r=r to s; a sv;:^r *r^ 



-* precede paragraph, we y t-anrfo, (!) by 

 means of the substitution 



where f is the constant 



f 



Ja 



The differential equation then becomes 



3? 



where 



W (*) and j(s) being the functions (^) 1/4 and - respectively, expressed as functions 



of a. In virtue of our hypotheses, it will be clear that q is a continuous function of s. 

 Corresponding to the interval a ^ x ^ b we have the interval 2= s == TT, and to each 

 pair of boundary conditions B, a B, B, JB for the former interval there corresponds a 

 pair for the latter ; these we shall denote by B', B', 'B', ^B' respectively. The 

 ivader will find that the pair of boundary conditions B' is 



~-h'u = at s= 

 <h 



where hf and H' are real constants whose values depend upon h and H respectively ; 

 he will also find that the pair 1& is 



u = at .s- = 0' 



u = ,, s = n 

 The pairs of boundary conditions B', *B' will then be obvious. 



