Inclined Plane when subjected to alternations of Temperature. 107 



Subtracting (a — £), (a — f 2 ), («— SJ, (a— B 2 ) from these 

 equations respectively, 



f ., w(l+\0 sin($ — , kvAj *m 



a ~ fl= 1~H + aEco.7 l—toj<—&> 



We) 



^E cos 



A „_f >/ w(l+H)sin(0+Q , m,L !W ^ 



Now the values of x represented by £ 1? £„ H 1? S 2 are those for 

 which Aa? = Aa? 1 = Aa? 2 = AX 1 = AX 2 . Therefore, by equations 

 (1), (2), (3), (4), 



Substituting the values of / t , ? v .fvfp an( ^ transposing, 





E\^ cos 



?(1+Xy sin(6— 



-i 1 



EAi cos 



fl — tt« = 



w(l — Xtf 2 ) sin (0 + 

 E\^ cos 



(7) 



1— w(l + Xy sin (0 + 

 w _ EX£ 2 cos 



a 2 ""~ w(l— Xfjj sin (0-0 ' J 



Eliminating* (fl — ?J, (« — ? 2 ), (a — Si), (a— E 2 ), between 

 equations (6) and (7), 



* It is easily seen from this elimination that the dilatation of the whole 

 plate is equal to one half what that of the part of it which dilates would 

 have been if that part had dilated freely. And so of th fi contraction. 



