WIIvI^IAMSON : STRAINS IN OPTICAI, GLASS 215 



In the case of the sohd sphere it is also interesting to find 

 the elastic strains. These are got by substituting the values 

 of Pr and Pt in equations (i) and (2). This yields 



«h ra2 / I 5 X-j 



Tangential extension^ = —j-^ ~ |_^ - r^(^- + - jj 



Radial extension^ = ; r I ~r + r 



'5<to + ok) 



\3n 9k/J 



9k> 



The nature of these strains was predicted at the beginning 

 of this paper, and it can be seen that in general the suggestions 

 were correct. The radial extension is practically constant for 

 it is approximately true for most solids that 4f = e, which gives 



I 5 



— — ^ = o. 

 3n 9k 



On the other hand, the tangential extension is equal to the radial 

 at the center, but diminishes, passes through zero and becomes 

 a compression for larger values of r. 



The form of these expressions shows that the stress at any 

 point may be analyzed into a hydrostatic tension proportional 



to a^ — — combined with a shearing stress which causes radial 



elongation and tangential contraction proportional to i + — 



r^ 

 and I— 7~, respectively. 



III. Cylindrical tube. 



Since the analysis is exactly similar, only the bare results are 

 written for the two cases considered, namely: 

 (Ilia). Linear heating, on outside only. 



d^ hr_hai2 



dr 2k 2kv 



• The values of the constants e and f are here expressed in terms of k (the com- 

 pressibility modulus) and n (the rigidity modulus). e = 1 , f = . 



3n 9k 6n 9k 



