WILLIAMSON: STRAINS IN OPTICAL GLASS 211 



gential. A tension is considered as a positive and a thrust 

 as a negative stress. Then 



^ = ePr -f Pt-fPt + ad (i) 



and - - -f P, + e Pt-f Pt + ad (2) 



where 6 is the change in temperature from the initial condition 

 of no stress, a is the coefficient of linear expansion and e and f 

 are elastic constants for the substance. ^ 

 The condition of equilibrium shows that 



r dPr 



Elimination of p and Pt yields 



d-P, , dFr 2 a dd 



r + 4 = • — 



dr- dr (e-f) dr 



and hence 



,„ 2 ct r r^ 



i-^Pr = -J^^j r^e dr + Ci- + C, (4) 



where Ci and C2 are constants to be evaluated by the boundary 

 conditions. Pt is then evaluated from equation (3). 



II. Solid sphere with a temperature distribution which is sym- 

 metrical about the center. — This differs from the previous case 

 only in the evaluation of the integration constants. C2must 



dd dP 



vanish as -j- and -r~ must be zero at r = o. 

 dr dr 



Hence 



,.r^ 2 a. C r^ 



r'P,=-(^3i)Jr^<'dr + Q- (5) 



III. Cylindrical shell or tube with temperature distribution 

 symmetrical about the axis. — ^In this case we made the assumption 

 that planes perpendicular to the axis remained plane. Except 

 at the ends this is justified and in fact it was found to be com- 



I ^ f 



* Young's modulus and Poisson's ratio, for the substance in question are - and -. 



e e 



respectively . 



