P. W. Bridgman — Crystalline Cylinders. 275 



The z axis is the axis of trigonal symmetry ; the axis of digonal 

 symmetry is the origin of 0, and there is a plane of symmetry 

 through the trigonal axis at right angles to the digonal axis. 

 The digonal axis and the plane of symmetry are here inter- 

 changed in position as compared with Love. 



The conditions are more complicated than for cuhic and tetrag- 

 onal crystals in that the axial displacement does not separate itself 

 from the others, but enters all the equations. But the compli- 

 cation actually results in a simplification as far as the first approx- 

 imation to the solution goes, for it will be found that the first 

 approximation is given by the warping function alone, the radial 

 and circumferential displacements requiring no corrective terms. 

 This means that the most important difference between a trigonal 

 crystal and an isotropic solid is in the presence of the warping 

 term. 



In virtue of the symmetry relations we see that u z must be an 



IT 



odd function of and an even function of — . Furthermore 



27T 



it must have a period of -^-. The simplest trigonometric func- 



tiou satisfying these requirements is sin 30, and actual trial 

 shows that this does indeed give a solution. The first approx- 

 imation is 



\ u r = B'r-\-A'r- 1 



U g=0 



!„ t= [ A> . + AV .- + ^,.-.] 



sin 3d 



The first approximation involves only the constant o lb peculiar to 

 the crystal. This solution contains four arbitrary constants, 

 with which the boundary conditions may be satisfied. This case 

 is simple enough so that the explicit form of these constants may 

 be written out easily, if one cares to stop with the first approx- 

 imation. 



The simple form of the first approximation, only a warping 

 term, makes it feasible to go on and obtain the second approx- 

 imation. Symmetry considerations including the condition of 

 trigonal symmetry show that Uq is an odd function of and an odd 



function of — — , and u r is even in and even in — — . The 



simplest trigonometric functions which satisfy these conditions 

 are sin 60 and cos 60 respectively. Accordingly we assume as 

 the second approximation 



Am. Jouk. Sci.— Fourth Series, Vol. XLV, No. 268.— April, 1918. 

 20 



