594 
Proceedings of the Royal Society of Edinburgh. [Sess. 
This proves the second of the subsidiary results. The first may be 
proved in the same way, replacing V by V — c. 
Thus the lemma is proved. 
It need hardly be added that the reversal of the order of integration 
is still allowable when the lower limit of integration with respect to v is 
altered from zero to, say, b, where b< B. This follows in fact at once by 
subtracting the equation in which the limits are zero and b, from that in 
which the limits are zero and B. 
§ 7. Theorem. — If f(u) is monotone decreasing with zero as limit at 
infinity , then 
cos uv du = 0. 
For, by the preceding lemma, in which we put V = 0, 
jo v ’ t)dw = \1du\ ®F(«, v, t)dv , . 
• (1) 
where 
F (u, v, t ) = e~ kv2 f(u) cos uv du. 
Moreover, J*cfe;JjF(u, v, t)du exists, for 
(W t)du =f(g) sin - sin Q- V dy, 
J b J q J b V 
using the Second Theorem of the Mean ; therefore, using it a second time, 
the left-hand side is numerically 
C°2e- M dv, 
0 Jo 
which vanishes when b moves off to infinity. 
Hence, by (1), proceeding to the limit with B, and changing q into Q, 
\*dv\^{u) du = ^Jl d u\l'F(u, v, t)dv 
= kt J /“)**/* eos •'« dv = JJA® fl -^ Wdu < 
using the Second Theorem of the Mean twice. Hence the left-hand side is 
numerically L tt/(Q), and therefore the same is true of the limits of that 
integral, when t = 0. But, in any finite interval ( q , Q), f(u) is bounded and 
therefore summable. Hence by § 2, 
dvj® e~ kv ‘ H f(u) cos uv du = 0 ; 
therefore, by what was already proved, 
^Io!o dv jj F (“’ v > ^ du "/(Q)- 
