Solid Cylinders of Elliptic Section. 169 



increases. Also the value of s in (104), while varying with 

 7), is quite independent of the density or of Young's modulus. 

 Taking t; = '25, we deduce from (104) the following values 

 for s : — 



Table XVIII. 



Greatest Strain answering to Instability Velocity. 



b/a - 



■2. 



•4. 



•6. 



•8. 



1-0. 



(i/ays= 



■104 



•428 



•970 



1-644 



2-281 



In a material such as steel or good wrought-iron the strain 

 given by this table for a circular cylinder in which l=10a 

 would answer to a longitudinal load of some 3 tons per 

 square inch, and a slight ellipticity in the section reduces this 

 but slightly. Thus in circular, or nearly circular, cylinders 

 whose length is not decidedly greater than 10 times their 

 diameter, it would certainly be only prudent to consider 

 the magnitude of the stress-difference and greatest strain 

 before applying so rapid a rotation as the instability theory 

 allows. In cylinders in which b/a is as small as *2, instability 

 may be expected to arise under quite a slow rotation, and to 

 attempt to rotate such cylinders with the velocity allowed by 

 the elastic theories would be extremely rash. 



§ 48. As regards cylinders whose length is less than 8 or 

 10 times their greatest diameter, the results of the instability 

 theory are hardly likely to prove satisfactory ; but there can 

 be little doubt of the general truth of the conclusion the theory 

 leads to, viz. that the tendency to instability diminishes 

 rapidly as the ratio of length to diameter is reduced. On the 

 other hand, while the application of the results deduced from 

 our elastic equations is not legitimate in short cylinders unless 

 rj be zero, or at least very small, there is no reason to suppose 

 that the maximum stress-difference, or the greatest strain, 

 will be either very much greater or very much less than in 

 long cylinders under similar conditions. One of the strongest 

 reasons for this statement is derived from the comparison of 

 Tables I. and II. with Tables XIII. and XIV. According to 

 these tables the limiting speeds allowed by either elastic 

 theory are fairly similar for long cylinders and for thin disks, 

 and it seems most unlikely that any disproportionately large 

 difference will exist in cylinders of intermediate length. The 

 greatest difference between disks and long cylinders occurs 



