576 Proceedings of the Royal Society of Edinburgh. [Sess. 
we may write the equation (1) in the following form 
oo 
\ q g(u) sin uv{h{u) - s ra }du = 2 j g(u ) sin uv{a n cos nu 4- b n sin nu}du. 
n—m 
The inequality (2) shows that we may apply the lemma of § 15 to the 
infinite series on the right-hand side of this equation, bearing in mind that 
the individual integrals in this series continue to exist when we write 
infinity for Q. Thus we can assert that both sides of the last equation 
continue to exist when we write infinity for Q, and that the equation still 
holds when we put Q = oo . Now, when this has been done, the right-hand 
side becomes a series of functions of v, which converges boundedly for all 
values of B in the closed interval (0, B). Hence we may integrate term- 
by-term with respect to v between the limits 0 and B, and assert that 
J>J» sin uv{h(u) - s m }du = 2 J* dv\ q g(u) sin uv{a n cos nu + b n sin nu\du (3) 
n=7?i 
so that both sides exist and are equal. 
Again, since g(u)u/u is monotone decreasing in the interval (q, Q), and 
cos B u may evidently take the place of sin uv in the preceding argument, 
[ cos B u{h(u) - s m }du = 2 f cos B u{a n cos nu 4- b n sin nu}du , 
J q U n=mj q U 
and by the same argument as that used above, this equation has a definite 
meaning, and remains true when we write infinity for Q. 
But by a direct application of the extended form of the theorem quoted 
which holds for an infinite interval, the result just obtained is equally true 
if the factor cos B u is omitted. Subtracting these two equations, we get 
f sWo _ cos B u){h(u) - s m }du = 2 I 1 - cos B u)(a n cos nu + b n sin nu)du. 
J q U n~mj q U 1 
But this equation may obviously be written in the form 
17 du[g(u){h(u) - s m }J7 s i n uv dv\ = 2 \ q du[g(u){a n cos nu + & n sin nu}\^ sin uv dv^{ 4) 
n=m 
both sides accordingly existing and having equal values. 
Now, since / exists, the corresponding terms of the infinite 
J q Vj 
series on the right-hand sides of (3) and (4) have been already (§§ 13, 14) 
proved to be equal. Hence the right-hand sides of (3) and (4) are equal, 
and therefore the same is true of the left-hand sides. 
But, again, for any integral value of n less than m it is equally true, and 
has been shown in the same corollaries, that 
