THE EQUILIBRIUM OF ELASTIC SOLIDS. 47 



Case I. ; but the direction of the principal axes is different, as in this case they 

 are parallel and perpendicular to the radius. The dark bands seen by polarized 

 light will therefore be parallel and perpendicular to the plane of polarization, in- 

 stead of being inclined at an angle of 45, as in Case I. 



By substituting hi Equations (18) and (20), the values of p and q given in 

 (22) and (23), we find that when r = a j , 



o + 2 

 2 



+2 



?\ ~ n I " Vi^l "P*-! / i 5 \ t\ 



9ft 3m/ ' a? a? \9ju, 3m/ . 



.(25). 



When r = a ls :- = f o + 2-^^p + Q , , 



r 9/t\ a.'-a. 1 / 9m\ of -of / , , . 



l zb ;- 



(i i \ i /o / s 4. Q "X r 3 / 9 



I 1 \ 7 1 / ^(.t-j (*! "T 01*,, \ 7 tt-q / & 



__^_ _^ I I h ____________ I _ I _ I _ rt _______^^__ I __^_ I 



9/* 3m/ ' ttj 2 a a 2 \ 9/x 3m / 3 a, a 2 a \9^i 3m/ j 



From these equations it appears that the longitudinal compression of cylin- 

 dric tubes is proportional to the longitudinal pressure referred to unit of surface 

 when the lateral pressures are constant, so that for a given pressure the com- 

 pression is inversely as the sectional area of the tube. 



These equations may be simplified in the following cases : 



1. When the external and internal pressures are equal, or h l = h i . 



2. When the external pressure is to the internal pressure as the square of 

 the interior diameter is to that of the exterior diameter, or when a^h 1 = a^h.,. 



3. When the cylinder is solid, or when a a = 0. 



4. When the solid becomes an indefinitely extended plate with a cylindric 

 hole in it, or when a a becomes infinite. 



5. When pressure is applied only at the plane surfaces of the solid cylinder, 

 and the cylindric surface is prevented from expanding by being inclosed in a 



Sr 

 strong case, or when = 0. 



6. When pressure is applied to the cylindric surface, and the ends are 



Sx 



retained at an invariable distance, or when - = 0. 



oc 



