Inclined Plane when subjected to alternations of Temperature. 113 



or generally when 



1 E/\ sin 2$ cos i ,,q\ 



> w (1 ± t\) sin (<£+0 sin (<£ — ' ' 



In the case in which this condition is tio£ satisfied, or w/zew Me 

 m;Ao/<? jo/a fe, having no point mechanically fixed, dilates or contracts 

 by the supposed variation in its temperature, let A. x B l (fig. 2) be 

 what AB becomes when heated by t v Then, since X A dilates 

 as it would do if fixed at the bottom, and X B as it would do if 

 fixed at the top, substituting the values (14) and (15) of X A 

 and X B in the third and first of equations (11), 



x l sin <p cost L 4Hj cos 9 sin <p cos 1 J 



XB 1= i fl (l+XQ sin pA fi^ ^ & + ') -M*-') I ■ 



1 *■ v x/ sm 9 cos 1 L 4ej cos $ sin epeost J 



■•■ A,B t = .(l+Xoll- ^t/ + >)sintf- t ) \ . (19) 



v 17 L 2E sin 2(£ cost J 



Similarly, 



A 2 B 2 = <1-X/ 2 )|1+ 2Esm2( £ C0S , > • ( 2 °) 



If the plate be first subjected to an increase of temperature, 

 becoming A^p and then to a diminution, becoming A 2 B 2 , the 

 value of Aj Bj from the former of the above equations must be 

 substituted for a in the latter. 



We shall then have approximately, 



A^ 2 =<l + X0(l-Xg{l-^^^^=^). (21) 

 * a v 1,x 2/ L 4E 2 sm 2 2(/> cos- 1 J 



By every such heating and equal cooling the bar will therefore 

 experience an exceedingly small diminution of its entire length. 



V. 



The descent of the plate when subjected to an increase and then 

 to a decrease of temperature, supposing the whole to dilate and 

 contract. 



Observing that X B (fig. 2) dilates as it would do if X were 

 fixed, and substituting for a, in the first of equations (12), the 

 value of XB (equation (15), 



1 4 sin <p cos c L 2E sm 2p cos 1 J 



Phil. Mag. S. 4 Vol. 38. No. 253. Aug, 1869. I 



