335 



sn 



cos 



') /T x 

 '- =w(L -W.T) 



') 

 1 



COS 



.'. #{sin (0+z)-|-sin (<p i)}=Lsin (q> i 

 cos =L sin ((jj ) 



2 sin (j> cos / 



Fig. 2. 



2 tan ^> 



When contraction takes place the con- 

 verse of the above will be true. The 

 separating point X will be such, that the 

 force requisite to pull XB up the plane is 

 equal to that required to pull AX down 

 it. BX is obviously in this case equal to 

 AX in the other. 



Let X be the elongation per linear unit under any variation of 

 temperature; then the distance which the point B (see fig. 1) will 

 be made to descend by this elongation 

 =X.BX 



If we conceive the bar now to return to its former temperature, 

 contracting by the same amount (X) per linear unit ; then the point 

 B (fig. 2) will by this contraction be made to ascend through the 

 space 



,- (1) 



2 I tan </> J 



Total descent I of B by elongation and contraction is therefore 

 determined by the equation 



j T , tan i / o \ 



t = Li\- (*) 



To determine the pressure upon a nail, %' ' 



driven through the rod at any point P 

 fastening it to the plane. 



It is evident that in the act of exten- 

 sion the part BP of the rod will descend 

 the plane and the part AP ascend ; and 



VOL. VII. 2 L 



