64 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



ference of retardation of the oppositely polarized rays of light is proportional 

 to the square of the radius r, or 



/= "bcjtav* = l)oi (q p) = boir -- , 



dp 



.-. = C r .: = 



Since if a be the radius of the cylinder, p = when r = a, 



P-f (<-*). 

 Hence q=^- (Sr'-a'). 



a 



By substituting these values of p and q in equations (19) and (20), and 

 d 8r d8r , ,, , 



r = Ifind ' 



c 4 being the temperature of the axis of the cylinder, and c, the coefficient of 

 linear expansion for glass. 



CASE XI. 



Heat is passing uniformly through the sides of a spherical vessel, such as 

 the ball of a thermometer, it is required to determine the mechanical state of 

 the sphere. As the methods ae nearly the same as in Case IX., it will be 

 sufficient to give the results, using the same notation. 



j dv c i 



r-j-sCj, .'. v = c t -, 

 dr r 



When &,=A, = the expression for p becomes 

 /2 IN" 1 /a a . ( a,'a,,' 1 0,0, 1 , a^-a,' 1 lc .\ 



7)=( - + - C,(0,-0,H i -- Hi? - TT^ Kf ...... ( 64 )' 



\9/i 3w/ i; \a,' - a,' r* a, - a r (,-*) (a, 1 - a/) J 



From this value of p the other quantities may be found, as in Case IX., 

 from the equations of Case IV. 



