Inclined Plane when subjected to alternations of Temperature. 115 



EX, 2 /? cos (j> 



1 2 sin <fi cos l 2w(l+\£ 1 )sin (0- 



X,B S 



EX 2 ^ cos <f> 



sin (</> — t) 



sin <£ cos 6 2w(l — X/ 2 ) sin (</> + t) 



Y . _j sin(<£ — t) EX 2 /? cos 



1 2 sin <£ cos l %w(\ + \t } ) sin (0 + t) ' 



EX 2 ?2 cos 



(25) 



X A = i a sin ^+0 

 1 2 2 sin cost 2w(l — \t q ) sin (</> — t) 



Adding the first and third of the above equations and the 

 second and fourth, and reducing, 



AiB, = «H- 



AoEc—a- 



EX 2 / 2 sin 20 cos i 



2w(l +\t 1 ) sin (<j) + i) sin (<j> — b) 



EX 2 / 2 sm20cos 



(26) 



2w(l — \t 2 ) sin (<£ + *) sin ($ — l) 



To determine the length of the plate when having been first 

 heated by tj it is cooled by t 9 , the value Afi x from the first of 

 the above equations must be substituted for a in the second, 



A 2 B 2 : 



a + 



EX 2 sin 2cj) cos c 



2w sin (<f> + 1) sin (<£ ■ 



OLi+x^ i-x/ 2 j 



or 



A 2 B 5 



a + 



(27) 



EX 2 (^ + ^)(^-/ 2 -X^g sin 2^> cost 

 2w(l+X^)(l — Xt 2 ) sin (</> + t) sm{(f> — i)' 



The bar will be lengthened if 



(t-Q^Xhh, or if (~- i^X. 



VII. 



When the plate is heated (tf), to determine what part is not di- 

 lated ; and when it is cooled (t°), what part is not contracted. 



x X (fig. 2) is the part which, when the plate is heated 

 (^ ), is not dilated; and x x Xj is the part which, when the 

 bar is cooled {t°), is not contracted. In the two cases the 

 points X and X x respectively may be considered points mechani- 

 cally fixed. Therefore taking XB to be represented by a in the 

 first of equations (7), and observing that a — | 1 = XB — X<^=B^, 



EX/ t cos 



Bx- 



w{\ +X/J sin(0 — i) 



Similarly, taking xA to be represented by a in the third of equa- 

 tions (7), and observing that a — H 1 = xA—xX— AX, 



12 



