236 Proceedings of the Roijal Irish Academy. 



where e is an indefinitely small positive quantity which has reference to 

 possible discontinuity, and ^ui, n^ are the initial and final values oi n: in the 

 present case we might make fxi = - M, fX2 = + ^li- 



It remains to be seen that the result given by this approximation is 

 correct. One method of expressing the argument for this is as follows : — 



Consider, first, the range of fx for which 



- 7r/2 + £ < arg. JUL < 7r/2 - e, 



where e is a given small positive quantity. In this range, the error in (13) 

 can be expressed as a fraction whose denominator, (the ratio of the denomi- 

 nator of (11) to its first term), is a function of /a, a, h, which tends to the limit 

 unity as h increases, and whose numerator is the sum of four terms 



^-(x (!f + a--2«) gfj.[u + x-2li) g-ij.[u-x-ir2b-2(i) g-fj. (a;-?< + 26-2-i) fl5^ 



each multiplied by a function of fj. which tends to a constant limit. As the 

 indices have real parts which are negative, and of the order of the product 

 of h and a finite quantity throughout, it is evident that the resulting error in 

 the double integral diminishes indefinitely as h increases. 



Consider, next, the ranges for which arg. jli lies within £ of ± 7r/2. The 

 error in (13) can now be expressed as a fraction whose numerator is of the 

 same form as before ; the denominator no longer tends to unity, but, however, 

 remains finite. By applying the second mean-value theorem to the real and 

 to the imaginary parts separately, it is readily seen that, if each of the expres- 

 sions (15) is multiplied by fx<^{u)du, and integrated from a to x, the resulting 

 integral is finite. On multiplying this by the product of a finite quantity and 

 djjilfx, the double integral through these ranges of fx is seen to be of order e, 

 which can be diminished without limit. 



Next, consider the portion of the contour to the left of the axis of imagi- 

 naries. Here, except when arg. jx is indefinitely near to 7r/2 or 37r/2, the most 

 important term in the numerator is 



- e-'^ (» - '^) F, (fx, a) . e'^ (^ - *) F^ (- fx, h), (16) 



and in the denominator the latter of the two terms is the more important, so 

 that when /x is great, the fraction is asymptotically equal to e'^(-'^-M). If we 

 substituted this asymptotic value, the integral along this portion of the contour 



would be 



-(p{cc-e)Aoge(ix\/n,), (17) 



where //i is the final value of /x ; and its argument thus exceeds that of ^i by 

 277. And this result may be justified, as was (14) in the former case. 

 Thus, by addition of (14), (17), the value of the integral (11) is 



- 27r*^(a.^-£). (18) 



