1892.] Mr Chree, On Long Rotating Circular Cylinders. 287 



such sections remain plane. Unfortunately this absence of curva- 

 ture could hardly be observed except at the ends, where our 

 solution cannot strictly be applied. 



The shortening of the cylinder per unit of length is given by 



( - $l/l) = ( _ w /z) = co 2 p (a 2 + a' 2 ) V /2E (9), 



with a' 2 = for the solid cylinder. Though it vanishes with r\, 

 this is in ordinary materials an important alteration. Its magni- 

 tude in terms of co 2 pa 2 /E — a quantity depending on the density, 

 Young's modulus and the velocity at the outer surface — is re- 

 corded in the following table for the value "25 of tj, for various 

 values of a'ja : — 



Table I. 



Shortening of cylinder per unit of length, <r\ — 25. 



a'ja= -2 -4 -6 '8 1 



( - 81/1) + (to 2 pa 2 /E) = -125 13 145 17 "205 -25 



The entry under a /a = applies also to the solid cylinder. 

 Since the shortening varies directly as ij, its amount in terms of 

 &pa 2 \E may be at once written down for any other value of rj. 

 Numerical measures of ( — 81/1) in two typical cases will be found 

 in Tables IX. and XI. 



§ 8. The other displacements of most interest are the altera- 

 tions 8a and 8a' in the radii of the two cylindrical surfaces. The 

 ratios of these alterations to the original lengths are given in 

 terms of co 2 pa 2 /E by the formulae 



(8a/a) -j- (co 2 pa 2 /E) = I {1 - v + (3 + y) a'*/a?} (10), 



{8a' /a') - {a?pa 2 /E) = £ {3 + V + (1 - v ) a? /a*] (11). 



Thus the radii of both surfaces are always increased. Taking 

 <a 2 pa 2 /E as constant, the following table shews how these altera- 

 tions of the radii vary with rj and with a /a: — 



Table II. 



Value of (8a/a) - (co 2 pa 2 /E). 



