The Hon. J. W, Strutt on BesseFs Functions, 341 



The compounding of particular solutions like (39) to repre- 

 sent given arbitrary values of (f) and -~ (or, in the beat pro- 

 blem, of (f> merely) presents no difficulty after what precedes. 

 Fourier bimself has given the complete solution for the case of 



sm 2 

 n = 0, when Jn+i(^) oc — r- ; but I am not aware that the pro- 



blem has been considered before in its generality. 



The problem appended forms a good example of the applica- 

 tion of BessePs functions to a special case, in which a numerical 

 result is required; it was not invented for the purpose, but pre- 

 sented itself in the course of an acoustical investigation two or 

 three years ago. At the time I was not able to give a solution. 



A rigid cylinder contains incompressible fluid which has been 

 once at rest^ and is set in motion in such a manner that at a 

 certain section (perpendicular to the axis) the velocity parallel 

 to the axis is expressed by 1 +f6r^, where r is the distance of any 

 point from the centre of the section. It is required to determine 

 the motion and the energy thereof. 



Taking the axis of the cylinder for that of z, and z~0 for the 

 plane section, while r = l is the equation of the cylinder, we have 

 the following conditions to which the potential (p is subject i— 



(1) that when ?-=!, — ~0 for all positive z ; 



(2) that when ^ = 0, -^=1 -f-yu-r^, from r = to r—l. 



dz 



The rate of total flow across <^ — is 



27Trdr{i + ^r^) = tt (1 + 1/^) . 



f'^ 



Jo 

 Let (pQ correspond to this distributed uniformly, so that 



-J5=H-1^ and ^o=(l+4/^)^. 

 For the remaining part of ^ we have 



Assume iov (j)^ 



where p is a root of the equation 



J'o(p)=0. 



Each term in the expression for </)^ satisfies Laplace's equation 

 and the condition laid down for the cylindrical boundary, while 

 it vanishes when z becomes infinitely great. 



