Mr Dixon, On a property of summable functions. 215 



and as before \f(x + 0)—f(a;)\<h over intervals E which are 

 together 



r= —n m—\ 



In this expression SlSS^,^ > ^l^r — (2w + 1) e 



>(6-a)-/3-(2w + l)e, 

 22er,m< 3a, 

 and the intervals E fall short by less than 



/8 + (2^ + 1) e + 6a + ^ (2/1 + 1) ^ 

 of the whole interval (a, 6). 

 The value of 



\ [f{x^-6)-f{x)Ydx<{h-a)h\ 



J E 



For the rest of (a, h) we have 



/{/(^ + 0) -f{x)Y dx^2 j{f(x)Y dx + 2J{f(x + 0)Y dx. 



To the first term of this, the set where \f{x)\ >nh contributes 

 a quantity < 27, and the rest a quantity 



< 2n-'h''{l3 + i2n + l)e + 6a+p{2n+l)0}; 

 treating the other term similarly, and putting 7 for w^A^/3, we have 



[ {f(x + d) -f{x)Y dx <(b-a)h' + 87 



+ ^n%'' [{2n + 1) e + 6a + ^ {2n + 1) 6]. 



fb 



Since now I {f{x)Ydx 



J a 



exists we may take 7 = /^^ thus fixing n. 



1 1 



We can also take a = — , e = 



n" w 



the latter condition fixing p. Then if ^ < -^ , the whole is less 



than a certain constant multiple of h^, and can be made as small 

 as we please by diminishing h. 



Thus 



Lt I {f{x + e)-f{x)Ydx = o, if f {f{x)Ydx 



exists, f{x) being a function which is summable in an interval 

 (a, b') which includes (a, b). 



VOL. XV. PT. III. 15 



