570 Proceedings of the Royal Society of Edinburgh. [Sess. 
Cor. — If f(u) = g(u)h(u), where J" | g(u) | du exists, and h(u) oscillates 
finitely at infinity, the same result holds. 
§ 12. Theorem 8. — If in the interval (q, oo ) the function f(u) is 
monotone decreasing, and approaches the unique limit zero at infinity, then 
[ dvf flu ) sin uv du = [ &^du 
Jo J q J q U 
provided the latter integral exists. 
Since sin uv du = (cos va— cos vh)/v, it follows, by the Second 
Theorem of the Mean, that if a<b and v>e>0, 
| \ b a f{u ) sin uv du | L 2f(a)/e. 
Hence, firstly , Jq f(u) cos uv du exists for each positive value of v (since, 
by hypothesis, f(u) tends to zero at infinity) ; secondly, it is, for all values 
of v greater than e, a bounded function of v ; and, thirdly, Jq f(u) sin uv du 
converges boundedly to its limit when A becomes infinite, for all such 
values of v. 
Hence we may integrate with respect to v between the finite posi- 
tive limits e and B, and we get, denoting for the moment the integrand 
f(u) sin uv by F(^, v), 
J>J q FK v)du = ffdv L t JqF(m, v)du= L t J7 dv^ F (u, v)du= Lt ^dufjF(u, v)dv 
A=x A=-jo A=x 
= f ^W(cos eu - cos Bu)du. 
J Q u 
Now, in the interval (Q, oo ), the function f(u)/u is positive, and the 
function (cos eu — cos B^) is numerically L 2. Therefore the right-hand 
side of the preceding equation is numerically L 2 J 1 K — } du, which, since by 
hypothesis f(u)/u can be integrated over the infinite interval, is a function 
of Q which has the unique limit zero at infinity. Denoting this function 
by 9(0,), we have, therefore, 
If dv \q F ( u ’ v ) du - d(0) L \*dv\*F(u, v)du L j*dv\® F(u, v)du + gSh. 
Let e diminish to zero in such a manner that the repeated integral in the 
middle of this inequality has an unique limit, say I. We thus get 
Jo dvfl F (u, v)du-g( Q) LI L ^do^fFu, v)du + g( Q), 
or, which is the same thing, 
\\du\l¥(u, v)dv-g( Q) LI L f®duj*F(u, v)dv + g( Q). 
Now let Q move off to infinity in such a way that the repeated integral 
