300 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 



The method is analogous to that whereby we obtain Euler's 

 formula for the greatest length permissible in a strut of given 

 section subjected to a given load. If instead of fixing the velocity 

 of the cylinder we suppose the ratio of length to diameter given 

 us along with the cross section, Professor Greenhill's relation 

 assigns a limiting speed which cannot be exceeded without ex- 

 posing the cylinder to buckling. 



The physical problem treated by Professor Greenhill is one of 

 great difficulty, and doubts may be entertained as to how closely 

 it is reproduced in the mathematical problem which he has solved. 

 This question would however lead us too far afield, and we shall 

 here consider merely the conclusions to which the instability 

 formulae lead, throwing all responsibility for the exactness of the 

 instability theory upon its author. 



§ 24. For the case of simple rotation Professor Greenhill 

 gives two formulae*. In deducing the first he takes 



at the ends of the cylinder, where y denotes the perpendicular 

 from a point on the axis, supposed bent under rotation, on the 

 straight line coinciding with its undisturbed position. This sup- 

 poses the axis constrained to retain its original direction at the 

 ends. The terminal conditions assumed in the second formula 

 are 



y 



" TO* 



d%*' 



This answers to zero curvature of the axis at the ends, or on 

 Professor Greenhill's interpretation of the problem to the vanishing 

 of the elastic bending couple. 



The conclusions to which the formulae lead appear at first 

 sight widely different, the length allowed by the first set of 

 terminal conditions in a cylinder of given section rotating with a 

 given speed being according to Professor Greenhill 9"46/7r times — 

 or approximately thrice — that allowed by the second. The differ- 

 ence is however in considerable part due to a slight slip made by 

 Professor Greenhill in attaching a numerical value to a quantity 

 he terms \id. For this he quotes (I.e. p. 199) the value 4"73, 

 whereas in reality 



\ld = 2-36502, 



or only half this value. 



* Those numbered (19) and (20) in Professor Greenhill's paper, pp. 199 and 200. 



