1M DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



. ; ?/ of noints which, together vnth its 



" 



obtained in 10-13 .how that, if any one of the canonical Stnrm- 



',,,h,, - ..* *, <*** -* > t8 



, , , ,,,,h,,K - .. 



,' 1 ' ,11 of the 1 .h normal fnnctions .Wjr . pa.r of honndary 

 Idi'til of the aune category (B', JBt. V. or ;B') as the fivst-ment.oned senes 



converge uniformly in the set. . 



17 As usual, let/(.) be any function which has a Lebesgue integral ^n (0, .). From 

 thLm I of the preceding paragraph it appears that, at a point .of the open interval 

 (0 ,) all canonical Sturm-Liouville series corresponding to /(.) have the same 

 limite'of indeterminacy as FOURIER'S cosine series corresponding to J (*) ; thee 



are therefore 



lim s n (). 



-* 



From the results obtained in 13, 14, it is evident that they are 



Now, if a is any positive number less than IT, we have 



f" f i , ,\ 8 n i (2w-l) < j, _ A 

 lim /, (s0 - snlt ' 



->-aoJa SlU^C 



by the last corollary of II., 6. It follows that, at a point s of the open interval 

 (0, IT), the limits of indeterminacy of the canonical Sturm-Liouville series corresponding 

 to/(s) are 



Recalling that a is arbitrarily small, we have thus established the theorem : t 



At any particular point of the open interval (0, IT) the limits of indeterminacy of 

 the canonical Sturm-Liouville series corresponding to an assigned function depend 

 only upon the values assumed by the function in an arbitrarily small neighbourhood of 

 the point. 



Let us next consider the case when s is an end point of (0, IT), It follows from 

 theorem III that the limits of indeterminacy of those canonical Sturm-Liouville 

 series whose normal functions do not all vanish at an end point are the values of 



lim 9, (s) 



*< 



* Since lim I Sn _i () = 0. 



* 



1 For the theorems of this paragraph cf. HOBSON'S paper cited in the Introduction. HOBSON, it will be 

 recalled, assumes that </ has limited total fluctuation in (0, -). 



