596 Proceedings of the Royal Society of Edinburgh. [Sess. 
has been shown to be a bounded function of (B, u, t), for the range of 
variables considered. 
By the same reasoning, we may introduce the limit with respect to t 
under the integral sign, whence using (1), in which we put B = oo , 
E* f dv [ e~ kvH f(u) sin uv du — [ 1 + 2 f e~ v ' 2 v cos ^ dv\ 
t -°J 0 J q J q U ~°\ J 0 Jkt J 
But, by the extended Riemann-Lebesgue theorem, the integral with 
respect to v last written down has, provided u is not zero, the unique limit 
zero when t approaches zero. Hence, when multiplied by the summable 
function f(u)/u and integrated from q to oo with respect to u, this integral 
disappears from our equation. Thus we get the required equation 
Lt f dv ( e' w /(a) sin uv du= ( &^du. 
0 J q ' J q u 
§ 9. Theorem. — If f (u) is expressible as the product of two factors , one 
of which, g(u), is monotone decreasing with zero as limit at infinity, while 
the other is of the form cos uV, then 
^Jo dv\ q e~ MH f(u) cos uv du = 0. 
By the lemma of § 6 we have 
Jo dvffer kvn g{u) cos u{y - Y )du = e~ hv2t g(u) cos u(y - N)dv 
= Q + 0)K' cos u{v - V )dv, 
using the Second Theorem of the Mean twice. Hence the first of these 
repeated integrals is numerically 
= < 7 (Q + 0)1 f - (sin ^(B' - Y) + sin uY)du I L 27r(7(Q + 0), 
1 J qu I 
L 2, say, 
where ^ has the unique limit zero when Q moves off to infinity. 
Hence 
| ^ dv j^e~ kv2t g(u) cos u(y - Y)du [ L z, . . . -(f) 
the integral on the left certainly existing, since, by the same argument as 
that used above, 
| l^dv^ e~ kvn g(u ) cos u(v - Y)du | = e~ km 2iTr g(q + 0), 
and therefore vanishes when b moves off to infinity. 
Again, g(u) is summable in the finite interval ( q , Q) ; therefore the same 
is true of g(u) sin uv and g(u) cos uv. Hence, considering, if we please. 
