Inclined Plane when subjected to alternations of Temperature. 109 



III. 



General solution of Cases I. and II., every part of the plate 

 being supposed to dilate or contract. 



Ill this case equations (1), (2), (3), (4) must be integrated 

 between the limits x and instead of between the limits x and 

 f „ £ 2 , Sp H 2 , the results of which integrations may be obtained 

 by making the latter quantities zero in equations (5). 



We thus get 



•i=(i+w,){ 



to sin ((f) — t) 

 2E cos 



aja, 



T 2Ecos</> J ' 



« 2 =(l->i 2 )| 



1 v iy \ 2Ecos$ J 



a /t > ,* fi ^sin (6 — t) r\ 



(11) 



a. 



Or, since ^ is an exceedingly small quantity, 



{.. wsmup—i) 1 

 1 2E cos <j> J 



f w sin (<£ + 1 



«»-«= -{/«*- 2Ecos<ft «>' 



A t -«= | 

 A 2 -a=-^ 



2E cos <p 



a >a, 

 2Ecos</> J ' 



(12) 



wsin (4> — i) 

 2 2Ecos<£ 



fFAe/i the plate (every part of which dilates or contracts) is heated 

 t° and then cooled t 2 °. 



By the heating the length a becomes a x or A x , according as 

 the fixed point is at the top or the bottom ; and by the cooling 

 these lengths become } a 2 and { A 2 . Therefore by the second and 

 fourth of equations (11), 



?#sin {4> + l) 



1 a 2 =(l— M 2 H 1 + 



a 1 iflj, 



2E cos cf> 

 Eliminating a x and Aj between these equations and the first 



