Inclined Plane when subjected to alternations of Temperature, 111 



upwards would equal that of X B to being thrust downwards 

 when the temperature is raised, an element at X will dilate 

 equally upwards and downwards, and the point X itself (sup- 

 posed the centre of the element) will remain fixed. 



In the same way, since \ x is a point so taken that the resist- 

 ance to X X A being pulled downwards is equal to that to XjB 

 being pulled upwards if the temperature is lowered, an element 

 at X x will contract equally upwards and downwards, and the 

 centre X x of that element will remain fixed. 



To determine the positions of X and X r 



Pressure necessary to thrust XA upwards = Kwf 2 XA, 



„ „ „ XB downwards =Kw/ 1 (a — XA), 



„ to pull XjA downwards =Kw/ 1 X 1 A, 



, „ X X B upwards =Kivf 2 (a — X X A) ; 

 Kivf 2 XA=Kwf 1 (a~TX), 



liivf^A. = Kwf^a—^A.) . 



Whence we obtain, substituting for/ x and/ 2 their values, 



sin (6— l) , f, tant~] ., .. 



XA =*«sm^oT;=^l 1 -fa^> • < M > 



XB =ia sm f + i) =ia|l+ ~\. . (15) 



z sin </> cost z L tan(jf>J v * 



XiK=ia s MpA =ha f l+ ^\. . { i 6) 



1 ; sin (f> cos i z L tan<£J v ' 



Xfi= ia ^^>=ia(l-P^\. . (17) 



- sin (p cos i * L tan <p J v ' 



X and Xj are therefore symmetrically placed in the bar. 



It is evident that while the plate is in the act of dilatation, the 

 point X may be considered mechanically fixed, and whilst it is 

 in the act of contraction, the point X x . 



The equal and opposite resistances at X and X x may, first, equal 

 or be less than the thrust of dilatation, in either of which cases 

 the whole plate will suffer dilatation or contraction ; or, secondly, 

 the equal and opposite resistances at X and X a may be greater 

 than the thrust of dilatation, in which case a portion only will 

 dilate or contract. 



Now the thrust with which the plate tends to dilate under an 



