292 Mr Chree, On Long Rotating Circular Cylinders. [Feb. 8, 



Class IV., (4 — n/7)/3 < v > 5. 



Sub-class (i), a /a < a^ja : 



f{r) vanishes over one surface within the cylinder, and the 

 radial strain is a compression within, an extension outside of this 

 surface. 



Transition case, a la = a-[\a : 



The radial strain vanishes over the outer surface of the 

 cylinder and elsewhere is a compression. 



Sub-class (ii), a /a > a{\a : 



The radial strain is everywhere a compression. 



§ 12. As illustrating Class II. and the transition to Class III. 

 we shall consider the values "25 and "3 of rj. The approximate 

 values of a 2 '/a in these two cases are respectively '4276 and 

 •3728*; i.e. the radial strain is everywhere a compression when, 

 7) being '25, a'ja exceeds "4276, and when, rj being "3, a'ja exceeds 

 "3728. When a'/a<a^/a the radial strain vanishes within the 

 cylinder over two surfaces whose radii r t and r 2 are the positive 

 roots of (16). The values of rja and rja for a series of values of 

 a'ja are given in the following table to three places of decimals: — 



Table IV. 



Radii of surfaces over which -r- = ; rj = "25 and "3. 



a'/a= -1 -2 -3 -4 



134 -273 -423 616 



fa= -966 -962 -947 -916 '839 



7?=25 K/a= -966 ; 



. A irja = l'528a'/a '154 315 "5 



v a \rja = 1-0 -993 "970 '9165 



The radial strain is a compression at points whose axial 

 distance lies between a' and r, or between r 2 and a, an extension 

 where the axial distance lies between r x and r 2 . 



§ 13. As illustrating Class III. we shall consider the value *4 

 of rj. For this we find approximately 



a; ja = -27735, aja = "3486. 



Thus when a /a is less than *27735 the radial strain vanishes 

 over only one surface within the cylinder, being a compression 

 within, an extension outside of this surface. When a'ja — "27735 

 the radial strain vanishes over two surfaces, but one of these is 

 the outer surface of the cylinder. W T hen a'/a lies between "27735 



* Equation (19) has the same roots when 77 = -3 as when t? = (4-\/7)/3. 



