1892.] Mr Ghree, On Long Rotating Circular Cylinders. 293 



and "3486 the radial strain vanishes over two distinct surfaces, of 

 radii r t and r 2 , within the cylinder, being an extension between 

 these surfaces, and elsewhere a compression. When a/a= *3486 

 these two surfaces coincide, so that the radial strain is nowhere 

 an extension. Lastly, when a'/a is greater than *3486 the radial 

 strain is everywhere a compression. The radii of the surface or 

 surfaces where the radial strain vanishes are given approximately 

 in the following table for the values 0, "1, "2 and '3 of a fa: — 



Table V. 



Radii of surfaces over which , = ; rj = 4. 



a'/a= -1 -2 -3 



rJa = V755a'/a 177 -364 -589 

 rja = -975 



§ 14. As illustrating Class IV. we shall consider the limiting 

 value o of i). For it we find approximately 



Oa'fa = -4472. 



Thus when a /a is Jess than "4472 the radial strain vanishes 

 over one surface, r = i\, within the cylinder, being a compression 

 within, an extension outside of this surface. When a'/a = "4472 

 the radial strain vanishes over the outer surface of the cylinder 

 and elsewhere is a compression. Lastly when a'/a exceeds *4472 

 the radial strain is everywhere a compression. The approximate 

 values of rja for the values 0, "1, "2, *3 and "4 of a /a are as 

 follows : — 



Table VI. 



Radius of surface over which -=- = ; rj = "5. 



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



rja = 2-4i4i9a'/a "244 "480 '704 -910 



§ 15. The expressions for the stresses call for but little 

 remark. In the solid cylinder »r vanishes over the cylindrical 

 surface, and the same is true of h> when i] = '5. Elsewhere these 

 stresses are always positive, i.e. tensions. The third principal 

 stress, «», vanishes everywhere when rj = ; but for other values 

 of r} it vanishes only over the cylindrical surface 



r = a/72, 



being a tension inside, a pressure outside of this surface. It is 

 easily shown that rr is never algebraically greater than w> nor 

 algebraically less than 7z. Thus at every point w> — 7z } = S, is a 



