19(5 DR. J. MERCER: STURM-LIOUVILLE SERIES OF NORMAL 



function of s, it foUows that the limits of indeterminacy of (39) at the corresponding 

 point of (a, 6) are 



Hence the theorem : 



At any particular point of the open interval (a, b) the limits of indeterminacy of the 



Sturm-Liourille series (39) depend only upon the values assumed by F (x) and | in an 



arbitrarily small neighbourhood of the point. 



In a similar way it may be shown that : 



At an end-point of (0, IT), where the functions (x) do not satisfy the boundary 

 condition v = 0, the limits of indeterminacy of the Sturm-Liouville series (39) depend 



only upon the values assumed by F (x) and 2 in an arbitrarily small neighbourhood to 



the right, or left, of the point, as the case may be. 



The statement of the theorem which corresponds to the third of 17 is left to the 

 reader. 



25. When F(a?) is of limited total fluctuation in an interval of (a, b), the function 

 /(*) is of limited total fluctuation in the corresponding interval of (0, ir) ; also, if x is 

 any point of the former interval, and s is the corresponding point of the latter, we 



have 



F(z-0)=/(s-0), F(*+0)=/( + 0). 



It follows at once, from the second theorem of 18, that : 



If F (x) has limited total fluctuation in an arbitrarily small neighbourhood of a 

 point x belonging to the open interval (a, b), the Sturm-Liouville series (39) converges, 

 and has the sum [F(z 0) + F(ic+0)] at this point. 



Again, we see from III., 24, that when fl (x) exists, w (s) exists and is equal to it. 

 Hence, from the third theorem of 18, we see that : 



At any point x of the open interval (a, b), where ft (x), the bilateral limit of F (x), 

 exists as a finite number, a sufficient condition that the Sturm-Liouville series (39) 

 may converge, and therefore have ft (x) for its sum, is that 



(x-y,) + F (x + y a ) - 2ft (x) 



y 



should have a Lebesgue integral with respect to y in an interval (0, a). 



The corollary to this theorem which corresponds to that of 18 is of interest, since 

 it may be stated without the intervention of the functions y, and y a . It will be found 

 to read as follows : 



At any point x of the open interval (a, b), where F(x-O) and F(a-+0) exist as 



