1892.] Mr Chree, On Long Rotating Circular Cylinders. 303 



Since l/a according to the formulae (26) to (29) varies as 

 (E/S)* or (1/s)* it is easy to adapt results such as those of 

 Table XV. to cases when values other than - 001 are allowed 

 to S/E or s. A caution may however be not unnecessary as 

 to the use of such results, viz. that unless r\ vanish or be very 

 small the solution obtained in the present paper is not altogether 

 trustworthy when l/a is markedly less than 10, and that the 

 Greenhill theory is probably in such a case still more untrust- 

 worthy. 



§ 28. Tables XIV. and XV. show that Professor Greenhill's 

 formula (24), assuming his theory trustworthy in itself, ought 

 not to be applied in determining the limiting speeds in hollow 

 cylinders whose length is less than 12 or 13 times their diameter, 

 without a check being applied by reference to the results of 

 the present paper. The thinner the walls of the cylinder the 

 more necessary is the check. In solid cylinders, a check of this 

 sort is less necessary, but it must be remembered that when 

 l/a is less than 10 the hypotheses on which (24) is based can 

 hardly be considered satisfactory. 



§ 29. While the exact law laid down by the Greenhill for- 

 mula, viz. that the limiting speed a>a as far as instability is con- 

 cerned varies as (a/If, cannot be relied on in short cylinders, 

 there can be little doubt that there is a continuous and rapid 

 diminution in the tendency to instability as l/a is reduced 

 below 10. On the other hand, a large increase in the limiting 

 speed allowed by the stress-difference and greatest strain theories 

 can hardly accompany this reduction of l/a. 



For while in short cylinders the strains and stresses will doubt- 

 less for ordinary values of tj vary appreciably with z, the mean 

 value of a certain strain or .stress-difference for a given value 

 of r when taken between + I seems hardly likely to vary much 

 with l/a. 



Thus it is a priori improbable that the greatest values 

 reached by these quantities in a short cylinder can be much 

 less than the values attained in a long cylinder, the values of 

 aa and a/a being the same, though conceivably in some cases 

 they may be appreciably greater (cf. § 31). 



The following considerations afford strong support to this 

 view : The expressions we have found for the maximum stress- 

 difference and greatest strain both in thin disks and long cylinders, 

 when coa, a /a, p and E are treated as constants, vary with the 

 value of 7j only within comparatively narrow limits (see § 17). 

 Unless then the influence of rj on the magnitude of these 

 quantities be very much greater in cylinders of intermediate 



