240 Proceedings of the Royal Irish Academy. 



hold within these limits, save in the immediate neighbourhoods of - oirj'I, 

 where the former fails, and of 7r/2, where the latter fails. 



First, suppose that r is neither a nor h. 



And first consider the portion of the contour to the right of the axis of 

 imaginaries. 



Making use of the approximate values, we see, as in the case of (11), that 

 for the portion of the contour for which arg. jx is finitely different for + 7r/2 

 in the fraction in the integrand, the second term of the first factor, and the 

 first term of the second in the numerator, and the first term of the denomi- 

 nator, are the more important ; and the approximate value of the integrand 



is accordingly 



7Ti{4:pryi(^'^p-'-'>p({)(p)dpdfi. (37) 



If we substituted this asymptotic value, the integral would be 



|%(r-e).log,W^O- (38) 



And this substitution may be justified, as follows, for example. So long 

 as arg. fx differs from ± 7r/2 by a finite quantity, the error in (37) is ultimately 

 of order ff^ ; the argument used in Part I., Art. 3, shows that this does not 

 affect the result. 



To the excluded values an argument much the same as that used in the 

 corresponding case in Art. 1 applies. The difference is that the functions of 

 fi which multiply the terms of type (15), (with u, x, replaced by p, r), are now 

 replaced by functions which tend to the form 



A + B/{^^p) + C/{^r). 



When multiplied by ixp(l>[p)d.p, and integrated with respect to p, the terms 

 involving B, C would give a finite integral, even if each were replaced by its 

 modulus. 



Thus the value of (34) along the first half of the contour is given by (38). 



But the multiplier of dfx in the integrand is a function of odd order in fx, 

 and thus, by similar reasoning, the integral along the remaining portion of the 

 contour is 



^^(r-a)log.(/,/^0, (39) 



where ix\ = fxic^'"'. 



Thus the total value of the integral is 



-7r^-0(r-O. (40) 



If r = a or h, the integral is evidently zero. 



Next, consider the integral derivable from (34) by changing the lower 

 limit of p into r and the upper into h. We may alter the integrand by 



