92 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Now deform C into the real axis indented at the zeros of S(^). Since 
F(^) is odd, the integrals along the straight parts of this contour cancel. 
The integral round the semicircle at O tends to zero with the radius, and 
the sum of the integrals round the other semicircles tends to 
But 
^ Ks‘^abV(Ks)Q{Ks)R{Ks) . 
(5) 
S(k){ J„'(K6)G,'(Ka) - Gn(Kb)J^\Ka} - P(/c)T(k) 
~ {k(X^G ^(^kci) — (^^)G^(k&) 
Hence (5) is equal to 
Q(/Cs)R(K:g) 
^S'(k,)T(k,)‘ • 
Again, consider the function 
J„(k6)G,'(k?^)} 
1 
K^ab 
( 6 ) 
-#>(0 = ^)«n(Af)G„(^f)G„(p^)G„(<rC) , 
(7) 
where the amplitudes of X, ju, p, and o- are 0 or tt, X + /x + p + o->0, and0^7i<J. 
The first term in the asymptotic expansion of 0(f) is 
cot — a)]e 
— 2nTTi-{-i^(K-\- p-\- ( t) 
W (Vpo-) 
Now, if C' is a path from O to A above the ^-axis. 
1 4>{0di= - f 1 <^a)du I , 
Jc' Jab Jok Jkb 
where KB is the line = N. But, if N->oo , / since X + p + p + o->0; 
hence 
r roo rco 
I ■“ I 4- ir))idrj i cji[iiq)id'q. 
Jc Jo Jo 
In the two integrals on the right of this equation put rj = u/(X + /x + p + o-) ; 
then 
f ^ )dv + - ^ )dv. 
Jc Xq-p, + p + (rJo V A + p + p + cr/ A + p-|-p + (T_/o \A-{- /X + p + O’/ 
It will now be shown that this is also true in some cases even when 
X + /x + p + cr<0. Assume that 
(i) amp A = 0, 
(ii) p + p + o- is negative and = ~y, 
(iii) the amplitudes of p, p, and a are 0 or tt, 
(iv) A + P + P + or = A — y>0. 
