276 P. W. Bridgman — Crystalline Cylinders. 



[ u r = Br+AV J + <#>,(/•) +/,(r) cos 60 

 u e = A{ r ) sin G( ? 



u z = /,(»■) sin 30, 



where 0, , f\ , and f t are second order terras, and f t is of the 

 first order. Substitution in the differential equations shows that 

 the assumption is justified, and gives differential equations for 

 0. • ,t\ i f\ > a,K l f% involving enough constants to allow the 

 boundary conditions to be satisfied. The solutions are as follows : 



A(r) = D,r 5 +D 3 r- 5 + D 3 r 7 + D t r~ 7 + B.r" 3 + B,r" x 

 /,(/•) = ai D,r 5 + a a D a »-~ 5 + a 3 D 3 r 7 + a.D.r" 7 +C 1 r" 8 + C,/" 1 

 /,(r) = A 1 r i + A,r-».+ A^ f - 1 « 



The constants A, B, A,, A 2 , D, , D 2 , D 3 , D 4 , are arbitrary 

 as far as the equations of equilibrium go, and may be so chosen 

 as to satisfy the boundary conditions. In any special case they 

 will involve the elastic constants and the dimensions of the spec- 

 imen. The constants Bj , B 2 , C, , C„ , a,, a 2 , a 3 , and a t on 

 the other hand are determined in terms of the elastic constants 

 only, being required to cancel the constant terms of certain non- 

 homogeneous equations, a, and a l have the especially simple 

 values -1 and + 1 respectively. 



Daring the process of solution of the equations it was most 

 interesting to watch how the equations for the unknown expo- 

 nents of r in f-fcr) and f£r) divested themselves of the elastic 

 constants, standing forth in the end as equations with numerical 

 coefficients, the solutions of which furthermore were integers. 

 The same thing was true for cubic and tetragonal crystals. 

 There may be a point here of some mathematical interest. 



The same general type of solution applies to those trigonal 

 crystals characterized by five instead of six elastic constants. 

 (See Love, p. 157.) It may be obtained from the six constant 

 case above by putting c 1B =0. The solution is of the same form, 

 but certain terms vanish ; <f>, vanishes, the terms in r~ 3 and r - ' 

 in f fi an d fi vanish, and the term in r~' in f % vanishes. 



The solution as given above is now in shape for numerical 

 computation. In the following will be given the values for 

 quartz of the dimensions used in the experiments. The inside 

 diameter of the cylinders was 1 and the outside diameter 5*5 on 

 an arbitrary scale. The following numerical solution differs 

 from the literal solution above in that the terms are added to 



