Solid Cylinders of Elliptic Section. 167 



recognizes may conceivably be very small with many forms 

 of support. There are obviously a series of other critical 

 velocities answering to 



2fil/ir = 2, 3,...i, .... 



where i is any positive integer. The larger the value of i 

 the greater would be the danger attending the bending of the 

 axis. Since /jl oc o>2, the corresponding angular velocities are as 

 the squares of successive integers. 



The previous considerations, it must be clearly understood, 

 are intended not to throw doubt on the existence of a species 

 of instability, such as Professor Greenhill imagines, but merely 

 to give a general idea of the uncertainties attaching to any 

 numerical details to which his theory leads on account of 

 possible divergences between the terminal conditions he 

 assumes and those existing in practice. The only positive 

 conclusion we have come to is that a tendency to instability 

 may be expected to show itself by a want of smoothness in 

 the motion and an undue wearing away of the inner edges of 

 the bearings. This tendency to instability might be seriously 

 increased by a slight departure of the centre of gravity of the 

 cylinder when at rest from its axis. 



§ 46. We have next to consider the nature of the hypotheses 

 by which the equation (95) is obtained. The assumptions that 

 the action of the " centrifugal " forces may be calculated by 

 collecting the mass of the cylinder into its axis, and that 

 the elastic stresses over a cross section give origin to a 

 couple proportional to the curvature of the axis, are certainly 

 not more exact, even near the centre of a long cylinder, than 

 the Bernoulli -Eulerian treatment of the flexure of a rod under 

 its own weight. Near the ends of the cylinder the strain and 

 stress must differ widely from that assumed by Professor 

 Greenhill, as he takes no account of the displacements which 

 exist in the absence of instability in a rotating cylinder. 

 These considerations show that while it is quite possible 

 Professor GreenhilFs formulae may lead to correct results for 

 short cylinders, there is no apparent reason from an elastic- 

 solid point of view why they should. My own view is that 

 the application of these formulae to cylinders whose length is 

 less than 8 or 10 times their diameter is certainly not 

 justifiable — an opinion in which I hope Professor Green- 

 hill will concur — and that in longer cylinders the application 

 of either formula can hardly be considered satisfactory 

 unless some definite evidence exists that the terminal con- 

 ditions it supposes are approximately satisfied. 



§ 47. Considering the uncertainty which prevails, it will 



