Solid Cylinders of Elliptic Section. 



163 



dB . 



When 7} is not very small it may be proved that -=— is nega- 

 tive along the whole minor axis when b/a is less than a certain 

 critical value b 3 /a depending on ij. The approximate values 

 of b 3 /a answering to the values *25, *3 and • 5 of rj are re- 

 spectively -523, -628, and -577. 



As b/a increases from b 3 /a the radial strain becomes an ex- 

 tension throughout a portion of the minor axis, and this 

 portion expands until eventually when rj > -3 it includes the 

 whole axis. The approximate values of y , the distance from 

 the centre of the points where the radial strain vanishes in the 

 minor axis, are given in the following table : — 



Table XVII.— Value of y /b. 



b/a. 



tj= -25. -3. -5. 



•6 



•633 — -963 



•8 



•909 -926 -507 



ro 



•966 ro — 



In the case of the first blank the radial strain is a com- 

 pression, in the case of the second an extension at every point 

 of the minor axis. When 77 has the values *25 and *3 the 

 radial strain is a compression in the length b—y , an extension 

 in the length y ; but when ?? = *5 it is an extension in the 

 length b—y , a compression in the length y . 



§ 43. As already indicated in § 38 an elastic theory of 

 rupture, such as the stress-difference or greatest-strain theory, 

 even if satisfactorily established, would not necessarily suffice 

 in the present case to fix the limiting speed. This failure 

 arises from the possible occurrence of instability, consisting 

 in a tendency in the axis of the cylinder to bend, it being 

 driven as it were by (i centrifugal " force from coincidence 

 with the straight line joining its ends. This question has 

 been investigated by Prof. Greenhill"*, who has deduced 

 formulae for the limiting speed in isotropic cylinders. Before 

 applying his formulas we shall briefly consider what is the 

 actual problem he has solved. He takes the cylinder, with 

 its axis displaced by rotation into a curved line, and supposes 



* Institution of Mechanical Engineers, Proceedings 1883, pp. 182 

 et seq. 



M2 



