FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 183 



at this point. It may be proved, from the formula 



that, at * = 0, these limits are 



sin 



and it will be found that, at s = IT, they are 



sin 



the numbers a being positive and arbitrarily small in each case. Hence we have the 

 theorem : 



At an end point of (0, IT) the limits of indeterminacy of all canonical Sturm- 

 Liouville series corresponding to an assigned function, save those whose normal 

 functions satisfy the boundary condition u = there, depend only upon the values 

 assumed by the function in an arbitrarily small neighbourhood to the right, or left, of 

 the point, as the case may be. 



Again, from theorem IV, it appears that, if one of the Fourier's series, say the 

 cosine series, corresponding to f (s) converges uniformly in an interval (y, 8) contained 

 within (0, TT), then all canonical Sturm-Liouville series corresponding to/(s) converge 

 uniformly in (y, S). It was shown in 13 that 



converges uniformly to zero for values of .s in (y, 8), as n increases indefinitely. Since 

 both the integrals 



8111 



converge uniformly to zero, for these values of s,f and since 



Ih^-Tnr 1 ** 14 (35) 



* This is clearly -?- I^.i (s). 



t In virtue of the first corollary of II, 9. 

 I This result follows from the fact that 



and that 



1 ['""l.O-l)^ = I ['[nYi'eo. rf\dl 

 irJ sin J/ Tj, r-i 



by the last corollary of II., 6. 



-. JT J, 



sn 



