434 
MATHEMATICS: W. B. FORD 
Theorem IV: Given any convergent improper integral of the form 
wherein \x p (x)\ for all positive (negative) values of x lying outside 
an arbitrarily small interval to the right (left) of the point ri: = 0 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<h) 
lim 
n — -{-co 
lim 
r i(x)p{n{x-a)\dx=j{a^^) 
k Jot 
'^£f{x)p[n{x-a)]dx=f{a~0), 
provided merely that / {x) satisfies suitable conditions in the neighbor- 
hood at the right (left) of the point. 
The proof of Theorem I follows directly from the fact (see for example, 
Dini's Serie di Fourier (Pisa, 1880), pp. 119-121) that if / {x) satisfies the 
indicated conditions there exists the general relation 
lim 
n= CO 
r ^ /' ^ ^ /(« -0) + /(a + 0) 
) J W ^ (», X- OL) dx = -'-^ ^ ^ J \ — — / , 
t/a 2 
whenever <p{nj t) is any function of the independent variables n and i 
satisfying the following three conditions, e always denoting an arbi- 
trarily small positive quantity: 
when — € < / < 0 
when 0 < ^ < € 
'(II) I (p {n, t) dt\ < Ci when —€</<€, Ci being a constant (de- 
pendent only on e) 
(III) \(p{n,t)\< Ci when |^<;^^^^^ ^^^|? ^2 being a constant 
(dependent only on c) 
Theorem II is a corollary of Theorem 1. 
Theorem III results from the fact (cf. Dini, I.e.) that if the conditions 
(I), (II), (III) above hold only for the positive (negative) values of t 
there specified, then, whenever / {x) satisfies suitable conditions at the 
right (left) of the point :r = a, we may write 
