﻿38 Dr. Dorothy Wrinch on the 



The radial and the transverse tensions have also their 

 maximum value on the axis, where they take the same 

 value, 



po) 2 (2\ + 3/x)/4a 2 (A + 2/x). 



The severest traction the cylinder is called upon to with- 

 stand is therefore 



po) 2 (2A, + 3yu-)/4a 2 (\+2yu), 



and it occurs radially and transversely on the axis of the 

 cylinder. 



In the case of the cylindrical annulus, the facts are entirely 

 different. The longitudinal tension reaches its maximum 

 value 



pco 2 X[{2Xi 3^a 2 /(\ + fi)-fjib 2 /{\ + fi)]mx^2fi) 



on the inner boundary. The transverse tension, in general, 

 reaches its maximum value 



po>*(2\ + 3/*) [a 2 + b 2 -2ab V(X + 2m[2X + 3/*]/4(A, + 2<a), 



on the cylinder r = r 2 . And the radial tension reaches its 

 maximum value 



f)co 2 (2\ + 3p){a-b) 2 mx + 2n), 



on the cylinder r=r 1 i As the ratio X//jl varies, the place at 

 which the maximum tension occurs varies. If the ratio b/a 

 is sufficiently large, i. e. if 



, /2\ + 3/x ,, . /a^+2/x 



b\f r t ->a, ba>\/— £-, 



V X+2[jl — ' ' ~ V 2X + 3/x 



the maximum transverse tension will occur on the outer 

 rim ; otherwise it will occur within the cylinder. Thus a 

 sufficiently thin annulus will have its maximum transverse 

 tension on the outer boundary. For different values of the 

 ratio X/fi, the maximum traction will occur transversely or 

 longitudinally, for a cylinder for which the ratio a/b is given. 

 And further, for a given ratio X/jjl, the maximum traction 

 will occur transversely or longitudinally. 



Higher Approximations for Expansion of a Solid 

 Infinite Cylinder. 



The first approximation for the radial extension was of the 

 form, 



p — r-t-q {a x r — r*/8) = r + qn x . 



