27$ 



1'. IT". B rich/man— Crystalline Cylinders. 



at the worst the boundary conditions are satisfied within 7% and 

 the equilibrium conditions within 10$. The approximation is 

 perhaps better than might be expected when one considers the 

 magnitude of the departure of the constants of quartz from the 

 condition of isotropy ; the terms in the equations which have 

 beeu treated as infinitesimals are not actually small at all. 



The numerical values of displacements, strains, and stresses 

 were next computed at various points in the crystal, and are 

 given in the following tables. The strains are those which cor- 

 respond exactly to the displacements, but the stresses Satisfy 

 exactly the boundary conditions and the equations of equilibrium, 

 and correspond only approximately to the displacements and 

 strains. 



If the substance were isotropic, there would be only two dis- 

 placements, u r , and u z proportional to s, only three strains e rri 

 e , and e„, and only three stresses rr, 80, and zz. The addi- 

 tional terms shown in the tables above are introduced by tbe 

 crystalline structure. It is curious that the extra displacements 

 occasioned by crystalline structure vanish at the inner surface ;, 

 there is here no warping, and all radii are shortened by the same 

 amount with no angular change. In general we should expect 

 the greatest extra displacement to be that given by the warping 

 function, because this is the only term in the first approximation. 

 As a matter of fact the warping is the greatest extra effect,, 

 rising at its maximum to 6'5% of the radial displacement at the 

 same point. If one assumed the substance isotropic with the 

 actual values of c n and c,„ the radial displacement at the inner 

 surface would be found to be — 2*32 + 10 -3 , which is close to the 

 actual value. Or if the displacement is computed for an iso- 

 tropic substance of the same cubic compressibility as quartz with 

 a Poissou ratio equal 0*25, the inner radial displacement will be 

 found to be — 2*43 + 10 -3 . It appears, then, that the crystalline 



Displacements. 



r 



1 



p-wr 



1 



p e 



1 

 p-« 2 



1 



-2-39 x 



10-°+ cos6# 



sin 60 



-6-97 x 10-* 2 + sin 30 



2 



-270 



" +1-87x10-* " 



-1-16 xlO" 5 " 



" + 1-56 x 10-"" 



3 



-346 



" +1-94 •' " 



-1-20 " 



+ 1-32 " 



4 



-4-34 



" +2-11 " 



-1-13 " 



+1-23 " 



5 



-5-27 



'' +2-48 " 



-0-78 " 



+ 1-32 " " 



5 '5 



-574 



+ 2-72 " 



-0-10 " 



+ 1-39 " " 



