P. IF. Bridgman — Crystalline Cylinders. 271 



One of the cubic axes is along the z axis ; the two others are 

 in the r-8 plane, one being the origin of 8 and the other at right 

 angles to it. The constants a, b, and c which appear above are 

 connected as follows with the elastic constants c n , c J3 , c ti . (See 

 Love, p. 157.) 



C. = a — c 



b + c 

 a — b 



+ c. 



-2 



The reason for writing the constants in this form is that for 

 an isotropic solid c becomes zero. In accordance with the 

 approximate method used here, c is to be regarded as an infini- 

 tesimal of the first order. 



If now the stresses above are expressed in terms of the dis- 

 placements and substituted in the equations of equilibrium, a 

 great simplification takes place. Because of the independence 

 of the solutions of z, e zz vanishes, and 8z and zr enter only the 

 third equation of equilibrium. Also u z enters only 8z and zr. 

 Now under such conditions the third equation of equilibrium is 

 satisfied if u z ~o. This also satisfies the boundary conditions in 

 2/', and because the solution is unique, it follows that u z must 

 actually vanish. 



That is, the deformation under external hydrostatic pressure 

 of an infinitely long hollow cylinder cut from a cubic crystal 

 with its axis along one of the cubic axes is one in which plane 

 cross sections remain plane. This is exactly true, without 

 approximation. We shall see that trigonal crystals behave quite 

 differently. 



We have left to consider now only u r and ug. These enter 

 into only rr, 88, and r8, and only the first two equations of 

 equilibrium. We now assume an approximate solution of the 

 form 



|w P = A 1 »' + A 1 r- 1 +/ 1 (r, 6) 



}ug = f,(r, 6) 



u r = k x r + A 3 r~ l is the solution for an isotropic substance, and 

 hence f^ and f t are to be regarded as infinitesimals of the same 

 order as c. The form of f t and f^ may now be guessed from 

 the symmetry relations. In the first place the elastic properties 

 have tetragonal symmetry about the z axis. The applied stress 

 system has circular symmetry about the z axis. The solution 

 must, therefore, repeat itself every 90°, and we expect a trigo- 

 nometric function of 4:8. Furthermore, the elastic properties 

 have digonal symmetry about the origin of 8. If the cylinder 



