224 Proceedings of the Royal Irish Academy. 



It might appear that this involves a mistake in sign, for superficially the 

 expression in brackets appears to change sign with X ; but in reality it does 

 not, being a function of even order in A. To put the matter slightly 

 differently, when in the former we write A = //c""', the expression alluded 

 to becomes 



Kn (- H^^p) Kn {e "^ fxa) - Z",, (e^~ fip) K„ {- ipa). (65) 



ISTow, K,i(e --i cc) is not identical with X„(e^o:); but, on the contrary, 

 they are connected by the relation 



— oTri Tri —iri 



Kn[e~''^oS) + K„{e^x) - 2i cos mr . {KnC~^ x) = 0, (66) 



and thus (65) is equivalent to 



Kn {iiJip]Kn (- ilia) - K„(- ifip) K,, {ijua]. 

 I consider, then, first the range of p from a to r. For this, the form (63) 

 is convenient. So long as the limits of A are expressed by - oo and + oo , the 

 two terms into which the left-hand member naturally breaks up cannot be 

 separated in integration, as the integrals so obtained do not converge. On 

 replacing, however, 



by }h ■ 



such a separation may be effected, and then the first term alone, including 

 the minus sign before it, is, by (16), equal to 



The second term may be shown to be zero. For the equation Kn^x) = has 

 no roots for which arg. x lies between ± 7r/2,* and thus Kn (- iXa) has no 

 zeroes for wdiich arg. A lies between zero and it. Consequently, in the second 

 term, the path of A may be deformed into a contour which is everywhere at 

 great distance from the origin. Along such a contour 



\Kn{-iXr)Kn{-iXp)Kn{ika)lKn{-i\a) (67) 



tends asymptotically to equality with 



for all arguments between and 7r/2, while between 7r/2 and tt there is to be 

 added a term in which the index is i\ (p + r), and in each case the fractional 

 error is of order A'^ Thus, by the reasoning of Arts. 3, 4, the second term is 

 zero. 



* MiicdonalJ, "' On Zeroes of the Bessel Function," § 7, Proc. Lend. Math. Soc, sxs. 



