1910-11.] Sommerfeld’s Form of Fourier s Repeated Integrals. 597 
g{u) as being zero when u is less than the positive quantity q, we have 
by §§ 2 and 4, 
L t [ dv f e~ kvH g(u ) cos u(y — V )du — f s * n v l L ( i u . . (2) 
t=0j 0 J q J q 'll 
From (1) and (2) we have 
Lt f dv C e-**g(u) cos u(v - V )du = + 6z, 
t=0j 0 J q J q 'll 
where |0| L 1. But the left-hand side of this equation is independent of 
Q, and the second term on the right has the unique limit zero, when Q 
moves off to infinity. Therefore, 
Lt f dv f e~ kvH g(u) cos u(v - Y)du = f ffW s ^ n u ^ du . . (3) 
t=0J 0 ./ q J q U 
Changing V into — V, we have, similarly, 
Lt f dv f e~ kvn g(u) cos u(v + Y)du = - [ ffdO s ^ n g u . . (4) 
t=oJ 0 J q J q It 
Adding the equations (3) and (4), we get the required result. 
§ 10. 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 sin uY, then 
J* dvj™ e~ MH f(u) cos uv du = 0, 
provided j S^du exists. 
By the lemma of § 6, 
Jo dxffe kvH g{u) sin u(y — V) du = ^du^e kv2t g(u) sin u(v - Y )dv 
= f du^—t [ e~ kv ' H u sin u(v - Y)dv. 
Jg U JO 
Now, 
jB e -hvH u s j n u ^ v _ Y)dv = [ - e kvH cos u(v - V)]® + 2 ktffve kvH cos u(v - Y )dv, 
which, as in § 8, is numerically less than 3. Hence the integral on the left 
converges boundedly to its limit \fe~ kvH u sin u(v — Y)dv, whatever values, 
fixed or varying, be ascribed to u and t. Similarly 
| [ B dvf e~ M g(u) sin u(v - Y)du I < 3e~ lb ‘‘ ( ^ ■ du, 
J b J q ' J q U 
