﻿

by Change in Angular Velocity. 3 



terms of a series of Bessel functions : 



«=i ' i'' o v a «y / Jo 



-j W,(«W«)«-"^ fWy,.^, . (5) 



a=] 



rJ (a„) 



f ${r)e va n* T l*dT, 



where J 1 (a )l )=0 and u n is the nth root of this Bessel 

 function of the first order. 



Taking </>(£) =Ii = constant and F(r) = 0, we have 



O, r n=1 ««J 2 («yi) 



(6) 



This is the solution for the case in which the water is 

 initially at rest and the cylinder suddenly rotates with a 

 constant angular velocity fl. The solution is given as 

 a problem in Gray & Mathew's ' Treatise on the Bessel 

 Functions ' (Ex. 38, p. 236, 1st edn.). 



Taking </>(£) = and F(r) = XI = constant, we find the 

 solution 



(j> _2c S Ji(* n r/c) H , C 2 



12 ?< w= i u n 2 {tx n ) 



This is the solution for the case in which the water is 

 initially rotating with a constant angular velocity O, and 

 the cylinder is suddenly stopped. The solution is given by 

 Stearne, Q.J. Math. xvii. p.' 90 (1881), and Tumlirz, Sitz. d. 

 k. Alcad. in Wien, lxxxv. (ii.) p. 105 (1882). 



The phenomena which (6) and (7) are supposed to 

 represent are at the basis of the formation and dissipation 

 of eddies by viscous action. To take one example, they 

 may be of importance in the theory of the aeroplane 

 compass. The experiments of Part I. were undertaken 

 to test the validity of these equations. 



§ 2. -Numerical Solution of Equation (7), and Discussion. 



The numerical solution of (7) is given in Table I. for the 

 values of r/c and vt/c 2 which are there indicated. The values 

 are probably accurate throughout to the fifth significant 

 figure. For small values of the arguments, the value 

 of <j>/Q sometimes differed from unity only after the sixth 

 figure. The values of J 1 (« /i r/c) for arguments greater 



B2 



