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Proceedings of the Koyal Society of Edinburgh. [Sess. 
Hence, as in the proof of the corresponding theorem in Section 3 (§ 13), 
letting, first, e diminish to zero in such a manner as to get an unique limit 
for the central integral, and then letting Q move off to infinity in such a way 
as to get an unique limit for the integral which appears in both the extreme 
members of this inequality, we see that these limits are equal, and hence 
that, however e and Q approach their limits, the two integrals have the 
same unique limit — that is, \*dv\*¥(u, v)du and \ q du^ J?(u, v)dv both 
exist and are equal. 
Hence 
rcc /*Q0 /-OO /»/ \ 
/ dv / f(u) cos uvdu= I ■— 1 sin B u du, 
J 0 J q J q U 
which, by the Second Theorem of the Mean, is numerically L 2f(q)/qB. 
Hence, letting B increase without limit, the required result follows. 
§ 24. Theorems 15 and 16 . — If in the interval (q, oo ) the function f(u) 
can he expressed as the product of two functions, one of which, g(u), is 
monotone decreasing with zero as unique limit at infinity, and the other, 
h(u), is of the form sin ku, or of the form cos ku, then 
§odv\ q f(u) cos uvdu = 0, 
provided in the former case J exists. 
These are, in fact, the results referred to at the end of the proofs of 
the corresponding theorems in Section 3 (§§ 13 and 14), as got by subtrac- 
tion and addition respectively of the results there added and subtracted. 
Cor. — When the upper limit of integration with respect to v is finite, 
we may reverse the order of integration. 
§ 25. Theorem 17. — If, in the interval (q, oo ), the function f(u) can 
he expressed as the product of two functions, one of which, g(u), is 
monotone decreasing with zero as limit at infinity, and the other, h(u), is 
any periodic function whose square is summahle, then, provided 
both exist, we have 
/: 
gW) 
du and 
/j 
f( u ) 
du 
Jo dv\ q f(u) cos uv du = 0. 
We shall, in the first instance, assume that the Fourier series of h(u) is 
free of constant term. 
Since the square of h(u) is summable, the Fourier coefficients of h(u), 
say a n and h n , are such that the series 'Lafn and 'Ehfn converge absolutely, 
by the lemma of § 16. 
Now, in any finite interval ( q , Q) of values of u, the function 
