296 Mr Ghree, On Long Rotating Circular Cylinders. [Feb. 8, 



§ 18. In comparing the results of Tables VII. and VIII. it 

 should be noticed that in a given material 8 = Es, if the two 

 theories really apply to all forms of strain, because in a bar under 

 simple longitudinal traction, s would be the longitudinal strain 

 answering to a traction S. 



It thus appears that the two theories are in exact agreement 

 when 7) = for all values of a'ja, and also when a' fa = 1 for all 

 values of tj. Also while differing in details they both lead to the 

 conclusion that as ij increases, other properties of the material 

 being supposed unaltered, the safe speed rises in a solid cylinder 

 but falls in a hollow cylinder for all values of a' fa; and further 

 that in a hollow cylinder of given material as a' '/a increases, a 

 remaining constant, there is a steady though not large fall in the 

 safe speed. To this last law the stress- difference theory recognises 

 an exception in the limiting case tj = "5, when the safe speed is 

 according to it the same for all values of a' fa. 



The most striking result is unquestionably the great fall in the 

 safe speed which according to both theories follows the removal of 

 a core however thin it may be, consistent of course with the mathe- 

 matical conditions for a free surface being satisfied. The magni- 

 tude of this fall is the more conspicuous the larger n is. Even for 

 7] = it amounts to over 29 per cent., and for t] = '5 it amounts on 

 the greatest strain theory to over 62 per cent. For tj = "25, 

 which is at least a fair approach to what is found in ordinary 

 isotropic materials, the mean of the falls in the safe speeds pre- 

 scribed by the two theories is approximately 38^- per cent. 



While the precise magnitude of the reduction in the safe speed 

 due to the removal of a thin axial core may be questioned by 

 those who regard with distrust existing theories of "rupture", the 

 fact that there is a large reduction must I think be admitted by 

 all who recognise the validity of the present solution, provided the 

 safe speed is really regulated by the elastic state of the material. 

 For our formulae show a large increase in the greatest values of 

 every stress and strain to follow the removal of a thin core ; so the 

 material can hardly fail to be brought considerably nearer to that 

 critical condition where the stress-strain relations cease to be 

 appreciably linear, whatever the precise elastic quantity may be 

 on which that condition depends. 



§ 19. To illustrate the use of the previous tables, and to give 

 an idea of the range of the numerical magnitudes of the several 

 quantities tabulated, I shall now consider some special cases. 



The first case, to which Table IX. refers, is designed to show 

 the magnitudes of the principal displacements and greatest strains 

 when the maximum stress-difference is of given magnitude. The 

 value of 77 in the material is taken to be "25. Since (— 81/1), 8a fa 



