330 On the Coefficients in BesseVs Functions. 



d 2 v/dz 2 = 0, and so the equation in r is not of the BessePs 

 type at all. This case I supposed a distinct one by itself and 

 always treated it as such (cf. Phil. Mag. Feb. 1886, p. 86, 

 a paper which, though published earlier, was written sub- 

 sequently to the one under discussion). It will I think be 

 more satisfactory to take the example which I actually applied 

 the solution to, viz. f(r) = Tr 3 , with T constant. For this 

 I found 



v = 8 (T//i) 2[{ Mkr) sinh kz}j{W3<£ka) cosh kl}] , 



the corresponding stresses being 



W?= - 8T2 [{ J 8 (Ar) sinh kz}/{PJ {ka) cosh Id}], 



z~d= 8T2[{Ji(ifcr) cosh kz}/k z J {ka) cosh kl], 



the former vanishing over ?* = a in virtue of 3 2 (ka) = 0. 



If I follow Mr. Bridgman, his point is that Tr 3 by itself 

 cannot be expanded in a Bessel series, but that Tr 3 + Br, 

 B being a constant determined by a simple integration which 

 Mr. Bridgman gives, is expressible in terms of the exact 

 Bessel's series which I supposed to represent Tr 3 alone. If 

 this should prove to be the case, then all that would be 

 necessary to make my solution complete would be to add to 

 the value for v the single term —V>rz/n, answering to a 

 torsional force of the simplest type. The same would be 

 true for the most general form of f(r), the value of B 

 varying with the function . If I rightly follow Mr. Bridgman, 

 the theorem he quotes is due to Dini, and amounts to accepting 

 r as a debased but necessary form of Ji(kr) — or in the general 

 case r l as a form of Ji(kr). — There is, Mr. Bridgman says, a 

 misprint in Dini's work and a misrepresentation of it in the 

 only text-book it appears in, thus some more definite con- 

 firmation appears desirable before finally accepting it. 



Dec. 19. — Since writing the above I have been able to 

 consult my friend Mr. Arthur Berry, and through him 

 Dr. E. W. Hobson, F.B.S. — who is a recognized authority 

 on Bessel Functions — on the point in pure mathematics 

 raised by Mr. Bridgman. These gentlemen substantiate the 

 accuracy of Mr. Bridgman's interpretation of Dini's result. 

 This means that in the problem treated by me in expanding 

 any function /(r) representing torsional forces, in addition to 

 a series of what are ordinarily recognized as Bessel functions 

 with arbitrary constants, there must be a term A r, of which 



