154 Mr. C. Chree on Rotating Elastic 



If gravity be supposed operative in aid of the restoration of 

 equilibrium, we should have to include in the boundary con- 

 dition relative to pressure a term gp£ in addition to T& 2 f ; so 

 that the more general result is obtainable by adding go/k 2 to 

 T. Thus 



-=-|( T +f> • • • • ( 38 > 



giving the rate of subsidence of waves upon the surface of a 

 highly viscous material. It could of course be more readily 

 obtained directly. 



When gravity operates alone, 



in= -Wk = -£k> w 



which agrees with a conclusion of Prof. Darwin*. A like 

 result may be obtained from equations given by Mr. Basset f. 



XVII. Rotating Elastic Solid Cylinders of Elliptic Section. 

 By C. Chree, M.A. t Felloiv of King's Collegej Cambridge^. 



Part II. — The Long Elliptic Cylinder. 



§ 35. "DY a long cylinder is here meant one whose length 

 -A3 21 bears to its greatest diameter 2a a ratio such as 

 is required for the legitimate application of Saint- Venant's 

 solution for beams. What this ratio may be depends on the 

 degree of accuracy aimed at, but the best authorities seem 

 satisfied with values of l/a which are not markedly less than 

 10. The cylinder is supposed to be rotating uniformly, and 

 to be free from all but " centrifugal " forces. In the paper 

 in the Quarterly Journal, already referred to, I obtained a 

 solution for a rotating elliptic cylinder, but its length was 

 supposed to be maintained constant by the application of suit- 

 able forces over the ends. This is a totally different case from 

 the present, in which the cylinder is supposed free from all sur- 

 face forces and capable of altering alike in length and diameter. 

 The present solution is thus completely new, except for the 

 case of a circular section which I have already treated else- 

 where §, and for the limiting value of rj when the alteration 



* Phil. Trans. 1879, p. 10. In equation (12) write ija*=k, and make 

 i=co . 



t Hydrodynamics, vol. ii. § 520, equations (21), (27). See also Tait, 

 Edinb. Proc. 1890, p. 110. 



X Communicated by the Author. 



§ Cambridge Philosophical Society's Proceedings, vol. vii. part vi. 



