Inclined Plane when subjected to alternations of Temperature. 103 



iv = weight in pounds of a portion of the bar 1 foot long and 



1 square inch in section. 

 i — inclination of the plane. 

 <£*= limiting angle of resistance between the surface of the bar 

 and the surface of the plane. 



f = 19 ~ ^ _ thrust per pound of the weight of the bar 

 Jl cos</> 



necessary to push it down the plane. 



£= ^ , L ' - = thrust per pound of the weight of the bar 



Jl coscf) l 



necessary to push it up the plane. 



x = distance in feet of any point P in the bar from the fixed 



end of it. 



o ^ 



fl s £ 

 I. "8 3 



<W S3 fl 



cc qj c3 



G9 03 



^! = what x becomes when T° becomes (T° + 1$ . 



a }> }> }> 



x „ }} (T°— if 2 ). 



X<2 — 



f 1= value of <2? in respect to the point where the 

 dilatation of the bar begins. 



f 9 = value of x in respect to the point where the 

 contraction of the bar begins. 



X 1 = what x becomes when T° becomes (T°+^). 



S3 o3 o 



"l +* 09 



^ r^3 



'. O +3 



i— i ^3 ^3 



A,= 



J.J 



a 



x 2 = 



;; 



X 



A 2 = 



)) 



a 



(T°-/ 2 ). 



S x = value of ,2? in respect to the point where the di- 

 latation of the bar begins. 



S 2 = value of 57 in respect to the point where the 

 contraction of the bar begins. 



* In the case in which there is a resistance of shear of the surfaces 

 as well as of friction; let \x. represent the unit of shear corresponding to a 

 unit of surface of 1 foot by 1 inch, and let cr be the area of the surfaces of 

 contact measured in the same units ; then \xv is the resistance of shearing 

 to the descent of the plate. Let also/ be the coefficient of friction, then is 

 fwer cos i the resistance of friction. If, therefore, tan (fi be the coefficient of 

 a friction equivalent to the actual resistance of friction and the resistance to 

 shearing, then 



w<r tan <j) cos L = iMT-\-fwcr cos i, 

 jxer sec i 



tan = 



+/• 



d') 



The value of (j) being determined by this equation, the following discus- 

 sio includes the case of adherence together with friction, and the resulting 

 formulae are applicable to that case also. 



