34 LORD llAYLEIGH. 



with tvvo similar équations; or witli use of (8) aiid (4) 



d /du , do , dio\ , „ dû 



[>■ -t- y-) 



if 



, , d /du , do , d/v\ , ., 



r = (3>. + 2f/.)x (7) 



One of the siniplest cases tliat eau be considcrod is that of a [jlate, 

 bounded by infinité planes parallel to xj/, and so lieated that Ô is a funo- 

 tion of c only. If^ further^ ô be symmetrical with respect to th(> iniddle 

 surface, the plate will remain unbent ; and if the m eau value of ^ be 

 zéro, the various plane sections will remain unextended. Assuming, there- 

 fore, that n, v vanish while w is variable, we get from {■]) and (4) 



^ = (/. + 2;.)|'-7^ = 0, (8) 



.P=Q=A^ — 7^, (9) 



S=T=U=0 (10) 



In (8) Ji is assumed to vanish, since no force is supposed lo act U[Hm 

 the faces, l'rom (S), (9) 



P=Q=_^';f- (ij) 



If the plate be examined in the polariscope by light traversiug it in 

 the direction of ^, the double refraction, depending upon the différence 

 between B and jP, of which the former is zéro, is represented simply 

 by (11). Dark bars will be seen at places where Ô = 0. If the direction 

 of the light be across the pJate, i. e. parallel to .~, there is no tendency 

 to double refraction, since every where P= Q. 



In the above example where every layer parallel to wj/ remains unex- 

 tended, the local altération of température produces its full eft'ect. But 

 iu gênerai the circumstauces are such that the plate is able to relieve 

 itself to a considérable extent. A uniforni élévation of température, for' 

 instance, would entail no stress. And again, a uniform température 

 gradient, such as would finally establish itself if the two surfaces of the 



I 



