JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. 9 APRIL 19, 1919 No. 8 



MATHEMATICS. — Straws due to temperature gradients, with 

 special reference to optical glass. Erskine D. Williamson, 

 Geophysical Laboratory, Carnegie Institution of Washington. 



In spite of the fact that the foundations of the mathematical 

 theory of this subject were laid as long ago as 1879,^ observers 

 have overlooked the simplicity of many of the results and their 

 bearing on practical problems. vSome time ago we had occasion 

 to evaluate the stresses that would occur in blocks of glass of 

 various shapes under varying heat treatments and in the follow- 

 ing pages we give the results obtained and some remarks on 

 their interpretation. 



To get a concrete idea of the nature of the problems, con- 

 sider a sphere of glass originally homogeneous which is being 

 heated at a uniform rate. Under this heating condition there 

 is set up a determinate temperature gradient such that the out- 

 side layers are hotter than those inside. If the sphere were 

 to consist of a number of detached spherical shells, these shells 

 would separate, but if the sphere be solid, internal stresses are 

 set up to counteract the separation effect. For mathematical 

 purposes we may evidently consider these two actions as taking 

 place separately, i.e., we may consider each element of the 

 sphere as undergoing an expansion due to the temperature 

 effect and then being brought back to its equilibrium position 

 by internal stresses. 



^ HoPKiNSON, J. Messenger of Math. 8: 168. 1879. 



209 



