210 Proceeding.'^ of the Royal Irish Academy. 



in the physical case), for negative vakies of a-, this arising from the circum- 

 stance that the integrals have to be evaluated after each element is multiplied 

 by cos \ct, where d is a constant (proportional to time). Taking, for example, 

 the integral (3), the portion involving g"'^'^ is now zero. For the portion 

 involving e'^'^, the range of u for which x + u- is positive contributes 



to obtain the contribution from the range from to - a;, we deform the 

 contour into one at infinity, but below the axis of real quantities, allowing 

 for the terms due to the poles thus passed over, and integrate along the new 

 contour, again using Fourier's theorem. Thus the value obtained for the 

 integral in (9) is now 



7r (7(oo) -t.S'(oo) . , . , ., .^T,^'('M-^'>5'(A) , 



2 6 (oo ) + ^*S (oo ) U[\)+ib{\) 



X e'^«0(2/)f/?i, (13) 



Sit denoting the sum of the residues. It seems unnecessary to go more fully 

 into the matter here. 



Arts. 3-6. A New Investigation of the Fourier-Bessel Integral Theorem. 

 Art. 3. The equation 



Lt r/^ 



XKn (- i\ r) d\ Kn{i\p) p(p (fj) dp = — ^ (r - e) , 



h J a ^ 



r > a, n unrestricted. 



Before proceeding to obtain the equation in Bessel functions analogous to 

 (9), (10), (12), it is desirable to make good a defect in the theory of the 

 ordinary integral theorem by extending it to the case in which the order of 

 the Bessel functions considered is algebraically < - 1. The equation, which 

 it is to be expected will hold then, is, of course, 



XdX f{\ r)J„[Xp)p(p{p)d,j = -^ (1 - e-'"") • (r - e) + ./, (r + e) } , ft < r < & 



C J -ao J O 



= l(l~e'"-')(f)(r-e), r = b 



= ^(1- 6-'"^') <^ ('' + £)> >' = « 

 = 0, r > b or < a, 



where the path of X is a contour on the upper side of the axis of real 

 quantities, and the Bessel functions have their principal values. 



It is, apparently, not legitimate to make b infinite without some restriction 



^■(14) 



