OiiR — Extensions of Fourier'' s and the Bess el- Fourier Theorems. 11 



Akt. 1. Minor Corrections. 



I WOULD first make certain minor corrections to my previous paper (E.I. A. 

 Proc, xxvii., A, No. 11), as follows : — 



p. 215, 1. 6. Insert 0(|o) as a factor in the integrand. 



p. 219, 1. 4. After " equation " add " unless arg. A = tt, when it is finite or 

 zero ". 



p. 219, 1. 26. As a foot-note read : — " When arg. A is very near tt, we must 

 use an expansion different from that which is ^'alid when arg. A is small ; 

 this may be done by introducing additional terms which involve e''^ as a 

 factor; these contribute nothing to the double integral." 



p. 220, 1. 15. Insert ri as a factor in the integrand. 



pp. 223, 225. In equations (63), (64), if we adhere to the contour at infinity, 

 we may add to it an infinite quadrant at each end and then halve the result. 

 The same is true of (69) with the second or the third form of the. left-hand 

 member. We thus obtain a closed contour, though the initial and the final 

 values of the integrand differ. Also conditions as to the zeros of the 

 denominator are irrelevant so long as the contour is kept at infinity. 



If we take the first form of the left-hand member of (69), when F{ix) is 

 '2fp{o'){d/dxy'K„(ix), F{-ix) should be '2fp(x){d/dxyKu{-ix) ; and, in 



the same form, if F{i\a) is K,n{;iXa), F{-iXa) should be {-)'"'" K,„ {- i\a), 

 while the derivatives referred to should be of both of these with respect to Xa. 

 (The point to be secured is that (66), as corrected below, should hold when Kn 

 is replaced by F.) 



p. 224. The factor i should be omitted from the last term of the left-hand 

 member of equation (66). 



p. 225, 1. 3 from foot. The value which is given should have a minus sign 

 prefixed. 



p. 230, 1. 5. It is unnecessary that the initial disturbance should be limited 

 externally. 



p. 231. In (76) for " A sin Xu - k cos Xtt" read " \ cos Xti + k sin Am ". 

 p. 242, 1. 21. If F,{Xa) is J„(\a) or aderivate, F^(Xa) is (-)'"-"/.„, (Aa) or 

 its corresponding derivate. 



p. 243. When n is negative, (52) cannot be obtained precisely as stated. 

 For, when a becomes zero, F2{Xa) will always dominate F^ (Art) if these are of 

 the type given in (45), (46). The equation may be regarded as a limiting case 

 of a somewhat more general theorem tlian that expressed by (49). Or it may 



