214 WILLIAMSON: STRAINS IN OPTICAL GLASS 



2« /hr^ hai^ \ Ci C2 

 ^' " ~(e-f) W + 6«r/ + 3 + r' 

 But Pr = o at r = a and r = ai so that Ci and C2 may be 

 evaluated, yielding 



^' i5«(e-f)L r ^ 



a^ + 5a%i2_5aj5 a'ai^ — 2a^ai^ + a^ai* 



a^ — ai^ (a^ — ai^)r^ 



] 



«n I o 5ai^ 



Pt = ,,...„ .^ [-2r^-^ + 



i5»c(e— f)L 2r 



a^ + 5a^ai^ — 6ai^ a^ai^ — 2a^ai^ + a^aiH 



a^ — ai^ 2(a^ — ai^)r' J 



(16). The values are obtained by assuming ai small enough 



to be negligible compared with a. 



ah I a-ai^\ 



i5K(e-f)\ r^ / 



ah. / a^ai^X 



Pt = 7 7^( -2r2 + a^ + -^) 



i5K(e-f)\ r^ / 



II. Solid sphere. Linear heating, on outside surface. 

 The treatment is exactly as in the previous case and the 

 resulting equations are : 



d^ _ hr 

 dr ~ 31c 



Pr = 7 h\-^' + a^l 



15K (e-f)L J 



Pt = —~ — ^\-2r' + a2 1 



15K (e-f)L J 



Except at the center these agree with case (16). At the center 

 in the case of a sphere we have a tension in all directions of 



a ha^ 



} TT but if there be a very small cavity the stresses must 



i5»c(e-f) ^ ^ 



be got from (16), and it is found that the radial stress vanishes 

 while the tangential tension is double the value for the solid 

 sphere. It follows that small cavities, due for example to bubbles, 

 make glass much more liable to breakage during heat treatment. 



