294 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 



correct measure of the stress-difference. Its greatest value, 8, the 

 maximum stress-difference, is found in the axis, being given by 



S = §a>>pa*{l-i V l(l- V )} (22). 



In the hollow cylinder it is obvious from (4) that rr vanishes 

 over both cylindrical surfaces and elsewhere is a tension. Also w 

 is always algebraically greater than 7? and so is necessarily a 

 tension, though when r; = "5 it may be vanishingly small. The 

 third principal stress 7z is never algebraically greater than m>. It 

 is a tension inside a pressure outside of the surface 



r 2 = (a 2 + a' 2 )/2, 



where rr is a maximum. The stress- difference may be w — rr or 

 H> — 7s according to the axial distance of the point considered, but 

 the maximum stress-difference is always the value of h, - rr at the 

 inner surface, and is given, since 7? is there zero, by 



B = $$r=a> = »> 2 {3 - 2 V + (1 - 2 V ) a"/a 2 } + 4 (1 - v ). . .(23). 



§ 16. In such a problem as the present the question perhaps 

 of most practical importance is the determination of the greatest 

 safe speed. Under ordinary conditions this is found according to 

 the stress-difference theory by attributing to 8 a limit found 

 experimentally ; on the greatest strain theory s is the quantity to 

 which an experimental limit is assigned. I have elsewhere* 

 discussed this question, pointing out that these theories at best 

 can do no more than indicate the limiting stress or strain con- 

 sistent with the stress-strain relations in the material remaining 

 linear, i.e. obeying Hooke's law. There is, however, no mathe- 

 matical objection to their application to determine a safe working 

 limit, provided this is consistent with the linearity of the stress- 

 strain relations. In the present case the question is complicated 

 by the possibility of the motion becoming unstable. It is 

 obvious, in fact, that in a long thin cylinder there is a danger 

 that the axis under rotation may cease to be straight and may 

 describe a spindle-shaped surface of revolution about the line 

 joining its ends. This has been pointed out by Professor Green- 

 hill -J", who has found formulae for the limiting speed, as depending 

 on this kind of instability, in terms of the material and dimensions 

 of the cylinder. 



I thus propose in the first place to give tables whence the 

 limiting speeds allowed by the stress-difference and greatest 

 strain theories may be obtained, illustrating their application to 

 special cases ; secondly, to give data showing how it may be deter- 



* Phil. Mag. September, 1891, pp. 239—242. 



t Institution of Mechanical Engineers, Proceedings, 1883, pp. 182 — 209, with 

 discussion, pp. 210 — 225. 



