FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 181 



Sturm-Liouville series corresponding to/() are identical with those of one of the four 

 series 



- sili Is P/(0 si" ^ tft + - si" \s ("/() sin * cfc 



77 Jo T Jo 



- COS A* ['/() COS l (/ + -COS \S f/(0 COS ft (/ 

 7T 'Jo 7T Jo 



+ ... + ? cos (n-1) ["/() cos (n-i)tfe+..., - (31) 



7T Jo 



- sin s f /"() sin tdt+- sin 2s ["/() sin 2t dt+ ... + - sin s f /(t) sin nt dt+ ..., (33) 



7T Jo' 7T Jo 7T Jo 



at each point of the closed interval (0, TT). It was then shown in 13-15 that, at 

 each point of the open interval (0, v), each of these four series has the same limits of 

 indeterminacy. We have therefore established the following theorem : 



I. At any point of the open interval (0, TT) each of the canonical Sturm- Lioueille 

 series corresponding to an assujned function which is integrable in (0, TT) in accordance 

 with LEBESGUE'S definition has the same limits of indeterminacy. 



In particular we have : 



II. If any one of the canonical Sturm-Liouville series corresponding/ to the function 

 converges at a point of the open interval (0, IT), then all of them converge at this point, 

 and all have the same sum. 



It was shown in 10, 12, that, at the end point s = 0, all canonical Sturm-Liouville 

 series whose normal functions satisfy B' have the same limits of indeterminacy as 

 (25), and that those whose normal functions satisfy "B ; have the same limits of 

 indeterminacy as (31) at this point. Then, in 15, we proved that (25) and (31) 

 have the same limits of indeterminacy at = 0. Since similar remarks apply to the 

 end point s = IT, we have the theorem : 



III. All those canonical Sturm-Liouville series corresponding to tlie function, whose 



s = 

 ,n, t -mal functions do not satisfy the boundary condition u = at s _ ^ have the same 



s = 

 limits of indeterminacy at 



S = IT 



In particular, the reader will observe that, if one of the series mentioned converges 



at an end point, all do so. 



Lastly, if the reader will examine the results obtained in 10-15, he will find that 



we have established the theorem : 



IV. If any one of the canonical Sturm-Liouville series corresponding to the 



