P. W. Briclgman — Crystalline Cylinders. 277 



give the correct longitudinal compression. This compression 

 was 23 = — —;— — i, and corresponds to a cylinder with closed 



ends, hydrostatic pressure being exerted entirely over the ends 

 as well as on the curved external surface. The constants c 13 and 

 c 33 both enter this additional term, so that the second approx- 

 imation, unlike the first, involves all the elastic constants. 



The elastic constants for quartz are taken from Love, p. 160, 



e n = 868, c, 2 = 70, c, 3 = 143 [ 



c 33 = 1074, c 44 = 582, c IB =-17lJ 



These constants are so chosen that a stress of 1000 kg/cm* is 

 regarded as unity. The values of the constants of the solution 

 are now as follows : 



A = - 1-388 X 10" S P D l = + 5-603 X 10" 9 P 



B = - 1-107 x 10" 3 P D 2 = + 3-623 X 10" 4 P 



A I= + 4-307 X 10" 7 P D, = -9-811 X 10"" P 



A 2 = -4-037 X 10" 4 P D 4 = -1-174 X 10" 4 P 



a,= -l, a 2 =+ 0-2985, o,= — 2-258, a,= 1, 



B= 8160A 2 , B 2 = - -06287A, 0,= - 0-1217A,, C 2 = + 0-0289A. 



With these values of the constants the displacements may be 

 found, then by differentiation of the strains, and then by substitut- 

 ing in the stress-strain equations the stresses. In this way the 

 stresses corresponding exactl}' to the given system of displace- 

 ments may be found. The goodness of the approximation is 

 now to be estimated from the closeness with which the exact 

 stresses satisfy the given boundary conditions and the equations 

 of equilibrium. 



The rr and r6 components of stress exactly satisfy the bound- 

 ary conditions on both exterior and interior curved surfaces, but 

 the component rz fails to satisfy the boundary conditions exactly. 

 At the external surface the effect is only 0*1$ of the applied 

 stress, but at the inner surface rz fails to vanish by a term fluc- 

 tuating in value, the trigonometric part of which is cos 39 sin 6#, 

 rising at the maximum to 1% of the applied stress. The approx- 

 imation to the equilibrium conditions was tested by direct differ- 

 entiation and substitution of the stresses, expressed as products 

 of trigonometric terms and polynomials in r. Tiie first two 

 equations of equilibrium are exactly satisfied; the third fails by 

 an amount which at the maximum is 10% of the largest terms 

 entering the equations of equilibrium. Hence one may say that 



