94 Proceedings of the Royal Society of Edinburgh. [Sess. 
^2inw 
— . — {G^(^a?) — respectively, and similarly for the other 
l/ir 
J„’s in Q(^) and R(0- Then, in the first case, 
p^irnr 
q(t)W) = - "^{Gn(ix)G„(iaei^) - G„(ta.)G„{ixe'’)} 
X (Gn(ir)Gn(iae^’) - G„(Ca)G„(^re<')}, 
and in the second case 
q(Om) = - ^-^{GMx)G„{(ae-i’) - Gn(C«)G„(f«-‘')} 
X {G„(Cr)G„(Cae-»') - G„(^a)G„(^re-*')}. 
Thus F(^) is replaced in each case by four functions of the same form 
as (p(0, to which formulae (8) and (9) apply. 
The integrals along the >y-axis will cancel one another. For example, 
consider the integrals arising from the products 
G„(C^)G„(^a.-)G„(Cr)G„(Cae^-^) 
and 
Gnaa)Gnaxe-^-)Gn{Ca)Gn{Cre-^-) . 
In the first we divide the arguments by x-\-r — 2a. In the second we 
first assume that 2a>x-\-r and divide the arguments by 2a — cc — r. We 
then make amp(2a — ;r — r) decrease from 0 to — tt, so that ultimately, 
when 2a — x — r<0, 
2a - X - r=- (x + r — 2a)e ~ . 
Thus 
2a-x-7’ x + r-2a' 
^xe 
2a- x-r x + 7' -2a' 
X-y 
remains unaltered, and therefore the integrals cancel. 
being even in (A — y), 
Accordingly 
the integrals on the right being all convergent. The restriction 
can now be removed. 
Again, if is the first term in the asymptotic expansion of F(^), the 
integral 
fmdi 
can be discussed in a similar manner. In order to avoid trouble regarding 
the convergency of the integrals at O, we first of all consider the integral 
