42 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



CASE I. 



The first equation is applicable to the case of a hollow cylinder, of which 

 the outer surface is fixed, while the inner surface is made to turn through 

 a small angle 86, by a couple whose moment is M. 



The twisting force M is resisted only by the elasticity of the solid, and 

 therefore the whole resistance, in every concentric cylindric surface, must be equal 

 to M. 



The resistance at any point, multiplied into the radius at which it acts, is 

 expressed by 



m 



rq, = T - 

 2 d'i 



Therefore for the whole cylindric surface 



j mirru = M. 

 dr 



M (I l 

 = - , -- 



Whence 80 



M fl i 



and m = 7TTsa ~ -- i 



2irbo6 Vet," a, 8 



The optical effect of the pressure of any point is expressed by 



T i Mb 



7=0,^ = 0,. ............................ (15). 



Tlierefore, if the soh'd be viewed by polarized light (transmitted parallel to 

 the axis), the difference of retardation of the oppositely polarized rays at any 

 point in the solid will be inversely proportional to the square of the distance from 

 the axis of the cylinder, and the planes of polarization of these rays will be 

 inclined 45 to the radius at that point. 



The general appearance is therefore a system of coloured rings arranged 

 oppositely to the rings in uniaxal crystals, the tints ascending in the scale as 

 they approach the centre, and the distance between the rings decreasing towards 

 the centre. The whole system is crossed by two dark bands inclined 45 to the 

 plane of primitive polarization, when the plane of the analysing plate is perpen- 

 dicular to that of the first polarizing plate. 



