70 Mr. C. Chree on Rotating Elastic 



so that the equation for ¥ is 



tan k f a tanh ha 



(32) 



Ma ka 



Again, for the solution involving the even functions, 



u = cosh kx + ~D cos k ! x, .... (33) 

 where 



cot Ida coth ka /0 . x 



~^r=— ~w (34) 



Equations (32), (34) give an infinite number of real values 

 for k\ and when these are known q, and n, follow from (29). 

 The most persistent motion (for which q is smallest) corre- 

 sponds to a small value of &, and to the even functions of (33). 

 In this case from (34) 



k'a — 7r, 27T, Sir, &c, 

 the first of which gives as the smallest value of q 



q = f i7r 2 /a 2 (35) 



The corresponding form for u is 



M = ^*-9'(l + cos(7ra?/a)) (36) 



This type of motion is represented by the arrows in the 

 following diagram : — 



1 z + 



On the other hand the smallest value of q under the head of 

 the odd functions is 



? = ^7r 2 (l-4303)7a 2 , (37) 



and the motion is of the type 



t ~ I 



♦ ^ t 



Terling Place, Witham. 



IX. Rotating Elastic Solid Cylinders of Elliptic Section. 

 By C. Cheee, M.A., Fellow of King's College, Cambridge*. 



Part 1. — The Short Elliptic Cylinder or Disk. 



IN the ' Quarterly Journal of . . . Mathematics,' vol. xxiii. 

 pp. 16-33, I considered various cases of isotropic elastic 

 solids rotating with uniform angular velocity about an axis 

 * Communicated by the Author. 



