1892.] Mr Chree, On Long Rotating Circular Cylinders. 301 



Amending this numerical coefficient, and employing as pre- 

 viously 21 for the length of the cylinder, in place of Professor 

 Greenhill's I, we may represent his two results under the forms 



co'p/Ek' 2 = (2-36502/0 4 (24), 



co'p/EK 2 = (7r/2iy (25), 



where k denotes the radius of gyration of the cross section about 

 a perpendicular through its centre to the plane of bending. The 

 rest of the notation has its previous signification. Professor 

 Greenhill apparently introduces no restriction as to the nature of 

 the cross section ; thus the formulae apply presumably to hollow 

 as well as solid cylinders. 



The terminal conditions assumed in (25) are based on the 

 elastic theory adopted by Professor Greenhill; while (21) merely 

 assumes the direction of the axis fixed at the ends, and so seems 

 exposed to fewer uncertainties. I shall thus direct my attention 

 to the first of the two formulae ; but results answering to the 

 second can be easily derived from those answering to the first by 

 replacing the factor (118251) 4 in equations (26) to (29) below by 

 (tt/4) 4 . 



§ 25. Let us then suppose that in a certain isotropic solid 

 circular cylinder the limiting velocity as prescribed by (24) is such 

 as to cause in the material a maximum stress-difference S. Then 

 remembering that a 2 /4 is the value of k 2 in a circle, and that the 

 angular velocity which appears in (24) is the same as appears in 

 (22), we immediately deduce 



8/E = (ri8251) 4 (ft/Z) 4 (3 - 4?) -5- 2 (1 - v ) (26). 



Similarly if 5 be the greatest strain answering to the velocity 

 prescribed by (24) we find by means of (12) 



s = (l-18251) 4 (ci/iy(Z-5 v ) + 2(l- v ) (27). 



In a circular annulus k 2 = (a 2 + a' 2 )/4. Thus, remembering 

 that s = ha'ja', we find from (23) and (11) for the maximum stress- 

 difference and greatest strain in a hollow cylinder, answering to 

 the limiting velocity prescribed by (24), the respective values : 



SfJS = (1-18251) 4 (a/iy (1 + a' 2 /a 2 ) x 



(3_2 7? + (l-27 7 ) a > 2 }-(l-7 ? ) ....(28), 



5 = (118251) 4 (a/iy (1 + a' 2 /a 2 ) [3 + V + (1 - v ) a 2 /a 2 } . . .(29). 



§ 26. It is obvious that the maximum stress-difference and 

 greatest strain answering to the limiting velocity prescribed by 

 the instability formula always diminish very rapidly as the ratio of 

 the length of the cylinder to its diameter increases. Also when 



24 2 



