1892.] Mr Ghree, On Long Rotating Circular Cylinders. 285 



♦Wp^ + a' + aVy^^^Xgl^ (5), 



J^VK + .'-^j^L. (6), 



-=0 (7). 



For a solid cylinder the displacements and stresses may be 

 correctly deduced from the above by omitting all the terms con- 

 taining a' 2 . This solution for a solid cylinder is identical with 

 one deduced from the case of a rotating spheroid* by supposing 

 the ratio of the axis of figure to the perpendicular axis to increase 

 indefinitely. 



§ 4. In what follows I shall assume and "5 as limiting 

 values of vj~. As regards the latter limit there is a difficulty that 

 will be best understood by reference to the relations 



n/m = 1 — 2v, 3n (1 — n/Sm) — E 



between E, rj and Thomson and Tait's elastic constants. 



If we regard E as finite, we must when n = h, have m 

 infinite. Thus those terms in the expressions for the stresses in 

 terms of the strains which contain m as a factor may remain 

 finite, though the corresponding strains vanish in the limit when 

 77 = ^. This explains the apparent inconsistency in the express- 

 ions supplied by our solution for this value of 77. The strains in 

 this case in the solid cylinder take the remarkably simple forms 



u = co 2 pa 2 r/8E, w = - co 2 pa 2 z/4<E, A = (8) ; 



so that the three principal strains, viz. -5- , u/r and -=- , are every- 

 where constant. The vanishing of A is a necessary consequence 

 of m being infinite, for this implies that the material is incom- 

 pressible. 



§ 5. The expressions for the strains and stresses in the axis 

 of a solid cylinder and at the inner surface of a hollow cylinder in 

 which a/a is infinitely small are, it will be noticed, totally 

 different ; for in the former case terms in « /2 /r 2 simply do not 

 exist, whereas in the latter case a' 2 /?' 2 = 1. There is thus, as in 

 the thin disk, a discontinuity in passing from a solid to a hollow 

 cylinder however small a/a may be. At first sight this appears 

 absurd, for it may be argued that if matter has a molecular 

 structure, as is generally supposed, then cavities exist everywhere 

 between the molecules, and there is no reason why a cylinder 



* Quarterly Journal..., vol. xxiii., 1889, p. 23, Equation (103). 

 t Phil. Mag. September 1891, pp. 235—6. 



23—2 



