114 MR JAMES CLERK MAXWELL ON THE 
tion of the oppositely polarized rays of light is proportional to the square of the 
radius 7, or 
2 dp 
P= Bien gig) —p)= : 
6, We =. (g—p)=b wr 
GD Sie wh Crees 
gel Pe ope tae 
Since if a be the radius of the cylinder, p=o when r=a, 
Bn CieoNes 
Pg t ,) 
Hence q= 4 (3 7? —a?) 
By substituting these values of p and q in equations (19) and (20), and making 
Devan) vei 
(63.) v= (satan) ets 
c, 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 are nearly the same as in Case IX., it will be sufficient 
to give the results, using the same notation. 

ee =H, goo 
Goes Parr 
aes 6, —9, 6, a,— 6, a, 
te 
Limes a, — 4, 
When 2, =/,=0 the expression for p becomes 
i} 58 x8 Bea 
(64.) p= (7 A ) Cr (6, -6,) { al pT —a, a” ea a \ 
9p 3m a,°—a,* 7? a,—a,r (a, —a,) (a,?—a,*) 
From this value of p the other quantities may be found, as in Case IX., from 
the equations of Case IV. 

Case XII. 
When a long beam is bent into the form of a closed circular ring (as in 
Case V.), all the pressures act either parallel or perpendicular to the direction of 
the length of the beam, so that if the beam were divided into planks, there would 
be no tendency of the planks to slide on one another. 
But when the beam does not form a closed circle, the planks into which it 
may be supposed to be divided will have a tendency to slide on one another, and 
