1910-11.] 
On Fourier’s Repeated Integral. 
577 
so that both sides exist and are equal. Hence, adding all such equations 
for integral values of n less than m, after multiplying by suitable coefficients, 
we have 
J du^ g(u)s m sin uv dv = Jo dv\ q g(u)s m sin uv du . 
Whence, expressing the fact already proved, that the left-hand sides of 
(3) and (4) are equal, and adding the equation so obtained to that last 
written down, we have 
f dv f f(u) sin uv du — f du [ f(u) sin uv dv = f ( 1 - cos B u)du. 
J o J q J q J 0 J q U 
Proceeding to the limit with B, we get, in this case, the required result, 
using the theorem of Riemann-Lebesgue (§ 10). 
Next, let be the constant term in the Fourier series of h(u). Then, 
by what has just been proved, 
/•" r 00 l 
I dv I g(u){h(u) - sin uv du= / -g(u) [h(u) — \af\du. 
J o J q J q y>' 
But, by Theorem 10, 
[ dv f g(u) sin uv du = f du . 
J 0 J q J q u 
Adding these two equations, the required result follows. 
Cor. 1 . — If h(u) is an odd function, the condition that 
J <i 
g(u) 
fe ^du exists 
u 
may be omitted. 
For, in this case, the coefficients a n are identically zero; therefore 
Theorem 9, § 13, is not used, while Theorem 10 does not require the 
condition in question. 
Cor. 2. — If h(u) is bounded in the interval (q, oo ), the condition that 
I - — I du should exist need not be explicitly mentioned in the enunciation. 
i q U 
For the condition is satisfied, since h(u) is bounded and / ^ du exists,, 
J q U 
g(u) being A 0. 
§ 18. It is clearly unnecessary for the argument in the preceding article 
that the square of h(u) should be summable, provided the series whose 
general term is ( | a n | + | b n | )/n is convergent. More generally, applying 
instead of Theorem 2 of my paper on the “ Integration of Fourier Series/' 
the theorem on which that theorem itself is based, namely Theorem 6 of the 
companion paper, ^ we may replace the integer n by any positive quantity 
* W. H. Young, “ On the Theory of the Application of Expansions to Definite Integrals/' 
presented to the London Mathematical Society. 
vol. xxxi. 37 
