
EQUILIBRIUM OF ELASTIC SOLIDS. 118 

dor _ =(5 
AF or — gi) o+p+a) +2 4 
0 1 
“= as ra) (o+p+q)+= —C,0 
The equation of equilibrium is 
d 
gq=ptrse . - (lL) 
and since the tube is supposed to be of a considerable length 
doa 
ae =e constant quantity. 
From these equations we find that 
1 1 dp 
Tae Sal Ne a 3a) (2p+7 7) 
egy 2 
9 38m 

and hence v=c, log r+c,, p may be found in terms of r. 
2 1 \ -1 1 
»=(52 +3n) ¢, ¢, logr+e, 75 + ¢, 
9u 3m 
H eee L we c one ae c (= wth 
ence qI= = +3) 1 €3 108 spat et yn +5n) C165 
= 2 L\i=t 1 
Since T= (g+p)=b4 (55 + gm) C e;—2 bwe, 
the rings seen in this case will differ from those described in Case III. only by 
the addition of a constant quantity. 
When no pressures act on the exterior and interior surfaces of the tube 
h,=h,=0, and 
il 2 = 2 ek 2 
oe (= +3a) s c, {log ro log a, Log a; 4% log a,—a, “Es | 


In 3m a," —a," a,”— a," 
a gu log a —loga, , a,7 log a,—a,” log a, } 
62. a= (55+ 3 om) C, ey { log r— ate + Fae +1 
=f 2 =u = 2 log a,—log a, 
raa(ue 2) ae, { 1— oo at aa \ 
There will, therefore, be no action on are light for the ring whose radius 
is 7 when 
a ae 
r2?=2—1 +. ae =z log 7 
CASE X. 
Sir Davin Brewster has observed (Edinburgh Transactions, vol. viii.), that 
when a solid cylinder of glass is suddenly heated at the cylindric surface a polar- 
izing force is developed, which is at any point proportional to the square of the 
distance from the axis of the cylinder ; that is to say, that the difference of retarda- 
