212 WIIvI^IAMSON : STRAINS IN OPTICAL GLASS 



pletely justified so far as experiment^ was possible. 



If the stresses be Pi, P2, and P3, parallel to the axis, radially, 

 and tangentially in the plane at right angles to the axis , respec- 

 tively, and X be the displacement parallel to the axis of the 

 element originally defined by the coordinates (x, r), the elastic 

 equations are: 



dX 



-T' = ePi — fPo — fPa + ad = constant 

 ox 



dp 



— =-fPi + eP.-fPa + ad 



- =-fPi-fP2 + ePs + ad 



dPo 



P3 = P2 + r 



f. 



dr 

 Pi rdr = o 



The last two equations are necessary for mechanical equiUbrium 



dX . 



and the substitution of a constant for ^r^ is the form taken by 



the assumption mentioned above. 



Elimination yields 



,dPi dd 



(e-f)^ + «^=o 



or (e-f) Pi + a^ = Ci (6) 



dP2 d^Po _ a dd 



^^^^ ^~dr+'dr^-~Je^)'dr 



yielding r^Ps = -^J rddr + -^ + C3 (7) 



The value of P3 is then obtained from 



dP2 



P3 = P.: + r^ . 



* The experiments bearing on this will be published in the series of papers on optical 

 glass now in the course of publication by this Laboratory. 



