40 LORD KAYLEIGH. 



lu (:i5), (36), (37) 



r(;preseiits tlie nieau temparature (above the standard) of tlie solid cyliu- 

 der of radius r. It is to be remarked that the double réfraction of the 

 ray at •;• is independent of the values of ù bevond r, and also of any 

 boundary-pressure. If ù increases (or decreases) continuously from the 

 centre outwards, the double refraction never vanishes, and no dark cir- 

 cle is seen in the polariscope. 



In the above solution if the cylinder is terminated by Hat faces, we 

 niust imagine suitable forces lî, given by (2S), to be operative over the 

 faces. The intégral of thèse forces may be reduced to zéro by allowing 

 a suitable expansion parallel to the axis. lîegarding div'ifl.: as a constant 

 (not necessarily zero)^ independent of r and z, we hâve in place of (28) 



The additions to P and Q, are '/, dw\d~, while {P — (d) reiuains uuchanged. 



If the cylinder is long relatively to its dianieter, the last state of 

 things may be supposed to reniain approximately unchanged, even 

 though the terminal faces be free from applied force. In the neighbour- 

 hood of the ends there will be local disturbances, requiring a more 

 elaborate analysis for their calculatiou, but the simple solution will 

 apply to the greater ])art of the length. 



The case of a thin plate wliose faces are every where free from applied 

 force is more diiîicult to tieat in a rigorous manner, but tlie following 

 is probably a sufficient account of the matter. By su])posing R= in 

 (3 S) we get 



I . N (f'i^^ , /'dit , «\ , ,^V 



(A + M^ = 7(5— A (^^+-;; 0^9) 



and using tins value of dwjdz, 



P = A'^r^'+-V V---;^. (40) 



A + 2/z \dr ^ rJ ^ "' ' dr ;.+2,v/ 

 ^'lix\dr ^ rJ ^ ' r A + 2/x 



A + 2w\r/r ' r/ ' ' r 



