MATHEMATICS: W. B. FORD 433 
those under which the Fourier series for / (a) converges) in the neigh- 
borhood of the point x = a. 
Theorem II: Given any convergent improper integral of the form 
I 
wherein {a) the function p {x) is even, i.e. p {—x) = p (x), and (b) the ex- 
pression l^t: p (x)\ for all values of x lying outside an arbitrarily small 
interval surrounding the origin remains less than a constant depending 
on the interval. 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) 
lint 
«= + 
provided merely that / (x) satisfies suitable conditions in the neighbor- 
hood of the point x = a. 
Theorem III : Let F {x) be any single valued function of the real vari- 
able X which, when considered for positive (negative) values only of x 
satisfies the following three conditions: 
(a)_ 
F (x) exists and = ^ 4: 0 ( Fix) exists and = — = 0 ) 
X = CO ^ ^ \X= — CO ^ ^ I 
(b) F{0)=0. 
(c) The derivative F^ (x) exists and is such that if we exclude the 
point a: = 0 by an arbitrarily small interval ( — €, e), (€>0), we shall have 
for all remaining positive (negative) values of x, \x F' {x)\<A^= an 
assignable constant depending only on e. 
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) 
lim 
lim 
= + °o k 
\ f{x)^F[n{x~a)]dx=j{a^Q) 
k Jot ax 
\ P/Wf F[«(^-a)]<ix=/(a-0)), 
k dX I 
provided merely that / (x) satisfies suitable conditions at the right (left) 
of the point x^a. 
