2IO WILLIAMSON: STRAINS IN OPTICAL GLASS 



A qualitative idea of the stresses can be immediately obtained. 

 Suppose the black parts in figure i represent concentric shells 

 which have become separated by the establishing of temperature 

 differences from shell to shell. To bring these together again 

 will require tensions radially in each of the white sections. The 

 radial tension must increase from the center to the outside as 

 each tension holds in the tension outside it. The effect will 



Fig. I. Concentric Shells. 



be radial stretching in all cases, but combined with the stretch- 

 ing in each shell there will be a total displacement inwards or 

 outwards for the outer and inner shells, respectively, and this 

 will result in a change of circumference which will be a decrease 

 for the outer shells and an increase for the inner. The super- 

 position of these effects gives the total strain. 



MATHEMATICAL DISCUSSION OF PARTICULAR CASES 



I. Spherical shell with a temperature distribution which is 

 symmetrical about the center. — Let the infinitesimal shell whose 



p 

 radius is r become finally of radius r -f- p so that - = the tan- 

 dp 

 gential (or circumferential) strain and — = the radial strain. 



Let Pp represent the radial stress at that point and Pt the tan- 



