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



(mUj 



§ 11. The variations in the sign of -=- in a hollow cylinder 



may be most easily investigated by means of the equation 



f(r) = (16), 



where f(r) is given by (146), regard being had to the sign of 

 the surface values of f(r), viz. : 



f(a') = - 2 v a* {(3 - V ) a 2 + (1 - S v ) a" 2 } (17), 



f(a) = - 2 v a 2 {(1 - 3t?) a 2 + (3 - v) «' 2 } (18). 



Let us denote by aj/a the least positive root, when real, of 



(x 2 + l) 2 (3 - 5 v f - \2x l (1 + v ) 2 (1 - 27?) (3 - 2 V ) = 0. . .(19), 



and by a t '/a the positive root of 



x 2 =(S v -l)f(S- v ) (20). 



Then a^ja is that value of a'ja for which f(r) has equal roots, 

 and a t 'ja is that value for which f(a) — 0. When a'ja<a^ja 



then f(a) — and so -j- at the outer surface — is positive. It is 



obvious that (a')~ 2 /(a') — and so -j- at the inner surface — is 



negative for all permissible values of t) greater than 0. For the 

 limiting case a'ja infinitely small, one root of (16) is of order a. 

 This root is given, neglecting higher powers of a'ja, by 



r> 2 = (a> 2 ) (1 + V ) (3 - 2 v )/(3 - 5 V ) (21). 



Though for shortness we refer to this case as that where 

 a'ja = 0, it is most convenient not to neglect the vanishingly 



small thickness i\ — a' within which -7- is negative, as we are 



thus enabled to include this case under the general classification. 

 The phenomena may then be grouped under four classes, according 

 to the value of 97, with transition cases. 



Class L, 77 = 0. 



Here (a')~ 2 f(a')=0=f(a), and f(r) is positive for all inter- 

 mediate values of r. Thus for all values of a'ja, the radial strain 

 vanishes over both surfaces of the cylinder, and elsewhere is an 

 extension. 



Class II., < rj < -3. 



Here a[ja is imaginary. 



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



