168 Mr. 0. Chree on Rotating Elastic 



suffice to indicate how these instability formulae may be 

 applied to check the application of the elastic theories of 

 rupture without entering on any elaborate calculations. 



In applying his first set of terminal conditions * Professor 

 Greenhill makes a numerical slip. His amended formula 

 for the limiting speed in a cylinder of length 21 is 



o>V/E*: 2 = (2-36502/Z) 4 .... (102) 



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

 an axis through its centroid perpendicular to the plane of 

 bending. The corresponding formula for the second set of 

 terminal conditions* is 



o>yE/c a = (w/2Q 4 (103) 



As we require the least velocity for which instability may 

 arise, k is the least radius of gyration obtainable, i. e. is b/2 in 

 an elliptic section. The corresponding plane of bending con- 



tains the minor axis. The terminal condition -~ z =0 of the 



second set of surface-conditions depends on Professor Green- 

 hill's elastic theory, which does not seem a close representa- 

 tion of matters near the ends. Thus, in selecting one of the 

 formulae for illustration I have preferred the first, as based 

 only on geometrical considerations. The results it leads to in 

 a cylinder of length L are, however, the same as the second 

 formula would give for a cylinder of length 7rL/(4*73). 



Taking then (102), suppose we determine by means of our 

 previous formulae the value of the maximum stress-difference 

 or greatest strain answering to the velocity which this formula 

 allows in a given cylinder. Then if this value of the stress- 

 difference or greatest strain be within the limits allowed by 

 the elastic theory of rupture, the greatest safe velocity is that 

 assigned by the instability formula, assuming of course that 

 the theory it is based on is satisfactory. Take, for instance, 

 the greatest-strain theory, and suppose our formula gives for 

 the greatest strain s 



ES/© a ,oa*=N, 



where N is a certain function of r) and b/a. Then ascribing to 

 o) the value given by (102), and putting K = b/2, we find 



5=(P18251) 4 (2&/a) 2 (a/Z) 4 N. . . . (104) 



This shows how very rapidly the greatest strain answering to 

 the limiting velocity of the instability theory diminishes as l/a 



* L.c pp. 198-200. 



