176 Lord Rayleigh on Stresses in Solid Bodies dice to unequal 



in which, however, /3 must vanish, if the cylinder is complete 

 through r = 0. From (34) 



*-*■+»-£&? £** ■■■■■ (35) 



and 



'-"j-s&tf*- *£**■}• • ■ • (87> 



It is on (P — Q) that the double refraction depends when 

 light traverses the cylinder in a direction parallel to its axis. 



In (35), (36), (37) 



% f f 0rdr 



represents the mean temperature (above the standard) of the 

 solid cylinder of radius r. It is to be remarked that the 

 double refraction of the ray at r is independent of the values 

 of 6 beyond r. and also of any boundary-pressure. If 6 

 increases (or decreases) continuously from the centre out- 

 wards, the double refraction never vanishes, and no dark 

 circle is seen in the polariscope. 



In the above solution if the cylinder is terminated by flat 

 faces, we must imagine suitable forces R, given by (28), to 

 be operative over the faces. The integral of these forces 

 may be reduced to zero by allowing a suitable expansion 

 parallel to the axis. Regarding dw/dz as a constant (not 

 necessarily zero), independent of r and z, we have in place 

 of (28) 



t, _ (du u\ . _ . dw „ , • 



The additions to P and Q are X dw/dz, while (P — Q) remains 

 unchanged. 



If the cylinder is long relatively to its diameter, the last 

 state of things may be supposed to remain approximately 

 unchanged, even though the terminal faces be free from 

 applied force. In the neighbourhood of the ends there will 

 be local disturbances, requiring a more elaborate analysis for 

 their calculation, but the simple solution will apply to the 

 greater part of the length. 



The case of a thin plate whose faces are everywhere free 

 from applied force is more difficult to treat in a rigorous 

 mnnner, but the following is probably a sufficient account of 



