80 Mr. C. Chree on Rotating Elastic 



§ 10. For a comparison of the results of the stress -differ- 

 ence and greatest- strain theories we have only to remember 

 that 8 = Es. We thus see that the limiting speeds allowed 

 by the two theories are the same when either 77 = or b/a = 0, 

 and that in any other case the limiting speed allowed by the 

 greatest- strain theory is the greater. The difference between 

 the two theories is the more conspicuous the larger the values 

 of 77 and b/a. For values of b/a other than the limiting- 

 speed according to the stress-difference theory falls as 77 

 increases, whereas according to the greatest-strain theory it 

 increases with 77. It is also noteworthy that while the 

 limiting speed on the stress-difference theory falls as b/a 

 increases, there is on the greatest-strain theory, unless 77 = 0, 

 a value of b/a less than 1 for which the limiting speed is a 

 minimum. This critical value of b/a is less the greater the 

 value of 77. 



§ 11. A considerable amount of caution must be observed 

 in applying the results of the tables to disks in which b/a is 

 very small. For, in the first place, no point in such a disk is 

 very far from the rim, where the accuracy of our results is 

 somewhat doubtful : and, in the second place, if the velocity in 

 such a disk were to alter, there would arise at every point a 

 reversed effective force proportional to the rate of change of 

 the velocity, directed approximately at right angles to the 

 major axis, and there would be a tendency for the elongated 

 disk to snap in two, just as if it were exposed to flexure in a 

 plane perpendicular to the axis of rotation. 



§12. When the rim is so nearly circular that we may 

 neglect the fourth power of the eccentricity e } we find from 

 (23) and (24) 



S=a)> 2 (3 + 77)/8, (25) 



s= cD*pa*(3 + <n) (I-71) {1+*V(1-*7)K£E. . (26) 



Thus when e 4 is negligible the maximum stress-difference is 

 the same as in a circular disk whose diameter equals the 

 major axis. Also, since 77/(1—77) cannot exceed 1, the greatest 

 strain, though greater than that in the circular disk, cannot 

 stand to it in a greater ratio than 1 + e 2 : 1. Thus, according 

 to either theory, the limiting speed is but little affected by 

 the substitution for a truly circular form of a slightly ellip- 

 tical. This constitutes a strong a priori probability that any 

 slight want of uniformity in the length of a disk's radius, 

 which does not remove the centre of gravity from the axis 

 of rotation, nor involve any sudden discontinuity in the 



