PROCEEDINGS 



OF THE 



Cambrifcrg* ^^iksop^iral Stodeig. 



Monday, Feb. 8, 1892. 

 Professor Darwin, President, in the Chair. 



Mr W. Heape, M.A., Trinity College, was elected a Fellow 

 of the Society. 



The following communication was made to the Society : 



Long Rotating Circular Cylinders. By C. Chree, M.A., Fellow 

 of King's College. 



§ 1. In Vol. vil, Pt IV., pp. 201 — 215 of the Proceedings I found 

 a solution for a thin elastic solid disk of isotropic material rotating 

 with uniform angular velocity about a perpendicular to its plane 

 through its centre. In the present paper the same method is 

 applied to a long right circular cylinder of isotropic material 

 rotating about its axis. The cross section of the cylinder when 

 solid is supposed of radius a, when hollow its outer and inner 

 boundaries are of radii a and a' respectively. The axis of the 

 cylinder is taken for axis of z, the origin being at the middle 

 point, and the notation for the displacements, strains, stresses, etc. 

 is the same as in my previous paper, except that the dilatation is 

 denoted by A. 



If Poisson's ratio, v, be zero the solution obtained here satisfies 

 all the internal and all the surface equations, whatever be the 

 ratio of the length, 21, of the cylinder to its diameter. But for 

 other values of n it is only true to the same degree of approxi- 

 mation as Saint- Venant's solution for beams, and like that solution 

 can legitimately be applied only when l/a is large. This re- 

 striction of Saint-Venant's solution, whether for torsion or flexure, 

 is not perhaps in general sufficiently recognised, but the best 

 authorities I believe regard it as sufficiently exact only when the 



VOL. VII. PT. VI. 23 



