,88 DR J. MERCER, STOBU-MOUVIIJ.E SERIES OF NORMAL 



.31 h.ve no difficulty in showing that a sufficient condition for this is that, as 



ft diminishes indefinitely, 



should converge uniformly to zero, for values of . in (y, 8), and for positive integral 



e us "suppose that the closed interval (y, 8) belongs to the set of points of (0, *) 

 at which /(.) ^continuous. It may be shown that the sufficient conation just stated 

 is satisfied when/(.) has limited total fluctuation in an interval (y , 8 ) enclosing 

 ( Y 8) in its interior. In doing so, we may evidently confine ourselves to the Case in 

 which /(.) is monotone ; for, in the most general case, /(,) is the difference of two 

 functions, each of which is monotone in (/, 8'), and has the points of (y, 8) for points 



of continuity. 



For values of ft which are not greater than a certain positive number ft we have 



]>->-/(.>] s ^r^ "< = [/<--/< .C sja W Lt * <w) 



at each point of (y, 8), where, as in 18, we have == ft < ft. The integral on the 

 right has been shown to be limited, for values of ft and ft, in (0, ft), and for all 

 positive integral values of n. Further, as ft diminishes indefinitely, /(-)-/(*) 

 converges uniformly to zero, for values of s in (y, 8) ; this follows from our hypothesis 

 as to the continuity of /(s). We conclude, therefore, that the integral on the left 

 of (37) converges to zero, as ft diminishes indefinitely, uniformly for values of s in 

 (y, 8), and for positive integral values of n. As the integral 



sm 



may be shown to have the same property, it will be now clear that we have established 

 the theorem : 



If the set of points at which f (s) is continuous includes a closed interval (y, 8) tying 

 within (0, ir), ami if f(s) has limited total fluctiiation in an interval (y', 8') enclosing 

 (y, 8) in its interior, then all the canonical Sturm- Liourille series cm-responding to 

 f(s) converge uniformly in (y, 8). 



Again, let us suppose that, for each value of s in (y, 8), 



has a Lebesgue integral with respect to t in an interval (0, ft). Then clearly 



We at once deduce the theorem : 



