Inclined Plane when subjected to alternations of Temperature. 10 



/. Aa?j=/,Aa?— 1 ~f 1 {a—x)Ax i 

 dx, . wL f . x 



Similarly, if the temperature T be reduced by t 2) and if the drag 

 upon the element Ax (in the act of contracting) of the portion of 

 the bar below it be applied as a strain after Ax has contracted 

 to / 2 A#j then this drag being represented, per square inch of 

 section, by {a—x)ivf 2i we have, as before, 



A« s =^A#+8(^.Aa;) j 



*f-4 + =5jfi(.-., ( 3 ) 



Case II. — TPften Me /ozm* end of the plate is fixed and the rest free. 

 Reasoning as before, 



m(l x Ax) , " v 



ES(/ 2 A,r) _ 



/ 2 A* 





"i/iv" «*7j 







AX,= 



-l x Ax 



-SftA*), 







AX 2 = 



= 4Aa? 



+ S(/ 9 Aa?), 







AX 1 = 



d x Ax- 



E [ 



x)Ax, 





AX 2 = 



= / 2 A# 



+ Wl ^{a- 



x) Ax, 





dX,_ 

 dx 



:/,-■ 



W \ U (a~z). 



? • • 



... (3) 



dX^_ 



dx 



^+."S yl (.-)- 



• • 



. . . (4) 







II. 







General solution of Cases I 



. and II., a part of the 



plate being 



supposed neither to dilate nor contract. 

 Integrating equations (1), (2), (3), and (4), and observing 

 that fj, f 2 , E l3 H 2 represent the values of x at the points where 

 dilatation and contraction respectively begin, 



