FUNCTIONS IN TIIK THEORY OF INTEGRAL EQUATIONS. 185 



*' -i-ies corren/M>n<fing In _/'(*) ////// -0,/ -,-///.. </,/,/ tlt.'refwe have <u (s) for their common 

 sum, is tli'it. us ft iliiiiiiiislti'* iii'f<-f!iiitf'/i/, 



f [/. (-0 + /. (* + ')-2 ()] ""ffilr 1 )* dt 

 o sin w^ 



uniformly to zero, for positive inteyral values of n. 



We proceed to verify that this condition is satisfied when /(.*) has limited total 

 fluctuation in an arbitrarily small neighbourhood (* a, + a) of the point s. Since a 

 function which has limited total fluctuation is the difference of two monotone 

 functions, we may confine ourselves to the case in which f(n) is monotone in the 

 neighbourhood. 



Observing that 



and that, for sufficiently small values of t, fi(st) may be replaced by f(st), it is 

 plainly sufficient to show that each of the integrals 



converges uniformly to zero, as y8 diminishes indefinitely. 

 For ft s a, the first integral is 



where, in accordance with the second mean value theorem, ^ /3 } ^ ft. It will 

 therefore have this property, if 



sn 



is limited, for values of /8 and y8 t in (0, a), and for positive integral values of n. Now 

 we have 



sin I / 

 Hence, by integration, 



= o _ 2 ' snm_ 2 "' 8n 



i/i = i m 



from which it is evident that the left-hand member is limited for the stated values 

 of /?, ft, and n. As similar remarks apply to the second integral, we have the 

 theorem : * 



//./'() has limited total fluctuation in an arbitrarily small neighbourhood of a 



* Obtained l.y HoBSON for the case in which q has limited total fluctuation in (0, *), ride the paper ciUxl 

 in thi 1 Introduction. The same remark applies to the corresponding theorems of 19, 20. 

 VOL. CCXI. A. 2 B 



