FUNCTIONS IN THE THEORY OF IMT.dKAL Kg CATIONS. 



Hitherto it has been assumed that the functions , (.r) satisfy the pair of boundary 

 conditions B. By using the results of 11, 12, and following the line of proof 

 indicated above, it will be found that the final result is unaffected when the pair of 

 boundary conditions happens to be either n B, *B, or B. Hence, as ar n (.s) is the sum 

 of the first n terms of the series (39), we have the theorem : 



The limits of indeterminacy of a Sturm-Liouville series at any point are the same 

 as the limits of indeterminacy of the canonical Sturm- lAouville series relcited to it at 

 the corresponding point of (0, TT). Further, if the former senes converges uniformly 

 in any set of points, the canonical series related to it converges uniformly in the 

 corresponding set of points oj ' (0, IT). 



24. The theorem of the preceding paragraph enables us to translate the theorems 

 <f ij 17-21 into theorems on the convergence of the Sturm-Liouville series 



*. (*) (" .</ (y) *. (y) F (y) dy+ % (x) f y (y) V 2 (y) F (y) dy 



J a la 



...... (39) 



As a preliminary we recall that, if s is the point of (0, IT) which corresponds to 

 x of (a, 6), 



It follows that the point s t of (0, IT) corresponds to xy, where, replacing t by y 

 for convenience of notation, y t is the function of x and y defined by 



fx //A i/a 

 (I) ' fa; 

 . i-y, \/ 



further, the point s + t of (0, IT) corresponds to the point x + y 3 , where 



The functions y lt y 3 evidently have positive values for positive values of y, and tend 

 to zero with y. 



Referring now to the results of 17, it will be evident that the limits of 

 indeterminacy of the series (41), at a point of the open interval (0, IT), are 



for values of a which are sufficiently small. Since f(s) is F (x) expressed as a 



2 c 2 



