432 
MATHEMATICS: W. B. FORD 
will have the property that 
^i-l /„(,)^/(«-0)+/(-+0). (2) 
An important special instance of an integral (1) having the property 
(2) is presented in the study of the convergence of the Fourier series 
for / (a), in which case the sum of the first n 1 terms of the series can 
be put into the form (1) with a = — ir, b = t and <p {n, x — a) = a cer- 
tain trigonometric expression. 
While the conditions upon (p(n, x-a) that will insure (2) have been 
extensively studied, especially by Du Bois Reymond, Dini, Hobson 
and Lebesgue, relatively little appears to have been done in the actual 
determination of such functions, the sole desideratum being the de- 
termination of the conditions themselves. In this connection the 
present paper would point out a noteworthy class of possible functions 
(p with special emphasis upon the corresponding integrals (1) to which 
they give rise. Four theorems are established, the second being especi- 
ally noteworthy in that it shows that to every convergent improper 
integral of the form 
^ 00 
p{x) dx = k::^ 0, 
wherein p (x) satisfies very simple conditions, there can be made to cor- 
respond a certain integral (1) having the property (2). 
The theorems are as follows: 
Theorem I : Let F (x) be any single valued fimction of the real variable 
X defined for all finite values of x and satisfying the following three 
conditions : 
(a) F (x) exists and = ^ 4: 0 
(b) F (-X) = -F (x). 
(c) The derivative F'(x) exists and is such that if we exclude the point 
x = 0 by an arbitrarily small interval ( — e, e), (€>0), we shall have 
for all remaining values of x, \x F' {x)\<A^ = an assignable constant 
depending only on €. Then, if / {x) be an arbitrary function of the real 
variable x defined throughout the interval (a, b), we shall have for any 
special value a (a<a<b) 
n=^-<^2kJa-^^^dx ^ ^ 2 
provided merely that / {x) satisfies suitable conditions (analogous to 
