1892] Mr Chree, On Long Rotating Circular Cylinders. 299 



Table XIII. 

 s = '001; £" = 20 x 10 s ; p = 7'5: v = 25. 



n S , oll f a'/a = Q 2 -4 -6 S 10 



Cylinder ' 



531 



pe^^condV 983 589 586 578 566 550 



In comparing the oo. 2 a of Table XIII. with the com of Table X. 

 it must be remembered that the values assigned to E in the two 

 cases are different. The proper basis of comparison may at once 

 be arrived at from the fact that the stress-difference and greatest 

 strain theories are in exact agreement in the limiting case 

 a/a = 1. 



§ 22. The results in the last three tables may easily be 

 applied to other special cases when 77 = - 25. Since ( — 81/1) and 

 Sa/a vary simply as s, if any value other than - 001 be assigned to 

 s we have only to alter the numbers in Table XI. in the same pro- 

 portion. We may also make use of the facts that $ varie3 

 directly as Es, and com varies as J Es/p, to adapt Tables XII. and 

 XIII. to other cases. 



For instance, let us take flint-glass, assuming for it 7) — '2o and 

 allowing to s the value "0008. Let us take E = 6 x 10 8 grammes 

 wt. per sq. cm., and p = 294 times the density of water, values 

 which are approximately the mean of those given by Professor 

 Everett*. Then to obtain numerical results for this case we have 

 only to multiply the values of (— 81/1) and 8a/a in Table XI. by - 8, 

 the values of S in Table XII. by (6/20) x '8 or "24, and the values 

 of com in Table XIII. by J(6/20) x "8 x (7\5/2-94) or '7825 

 approximately. 



§ 23. We have next to consider the limits which Professor 

 Greenhill has found for the safe speed when none but "centrifugal" 

 forces are supposed to act. He imagines the cylinder bent under 

 rotation so as to be in equilibrium under the "centrifugal" forces, 

 which tend to increase its bending, and the elastic forces whose 

 tendency is to keep it straight. He thence . arrives at a relation, 

 varying with the terminal conditions, between the velocity of 

 rotation and what is practically the ratio of the length to the 

 diameter of the cylinder. Taking the velocity and cross section as 

 given, Professor Greenhill regards this as fixing the greatest length 

 of cylinder in which rotation is stable. 



* Units and Physical Constants, 2nd Edition, Art. 64. 

 VOL. VII. PT. VI. 24 



