1892.] Mr Ghree, On Long Rotating Circular Cylinders. 291 



f(r) vanishes over two distinct surfaces within the cylinder. 

 Between these the radial strain is an extension, elsewhere it is 

 a compression. 



Transition case, a'ja = a 2 '/a : 



The two surfaces over which f(r) vanishes coincide. At this 

 surface the radial strain vanishes, elsewhere it is a compression. 



Sub-class (ii), a' /a > a 2 '/a : 



The radial strain is everywhere a compression. 



Transition case to Class III., rj = "3 : 



This follows the same laws as Class II., except that for a' /a = 

 one of the two surfaces over which f(r) vanishes coincides with 

 the outer surface of the cylinder, so that there is not, as in sub- 

 class (i) above, a volume of finite thickness at the outer surface 

 wherein the radial strain is a compression. 



Class III., - 3 < rj < (4 — j7)/3, i.e. -4514 approximately. 

 Here a-[\a is real. 



Sub-class (i), a fa < a t '/a : 



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' /a = a-^ja : 



f(r) vanishes over a second surface, but this coincides with 

 the outer surface of the cylinder. 



Sub-class (ii), a//a< a'/a< a 2 '/a : 



f(r) vanishes over two distinct surfaces within the cylinder. 

 Between these the radial strain is an extension, elsewhere it 

 is a compression. 



Transition case, a'ja = a 2 '/a : 



The two surfaces over which /'(>') vanishes coincide within 

 the cylinder. At this surface the radial strain vanishes, elsewhere 

 it is a compression. 



Sub-class (iii), a'ja > a 2 '/a : 

 The radial strain is everywhere a compression. 

 Transition case to Class IV. 

 7] = (4 — v7)/3 ; a t '/a = a 2 '/a = '3728 approximately. 



This follows the same laws as Class III., except that Sub-class 

 (ii) is not represented, and for a /a = '37 28 the radial strain 

 vanishes over two surfaces both coinciding with the outer surface 

 of the cylinder. 



