22 Sir W. Thomson on the Thermoelastic, Thermomagnetic, 



the dodecahedron in a determinate manner (having for its cor- 

 ners four of the eight trihedral corners of the dodecahedron) ; 

 and yet they may differ, and in all probability they do differ, 

 in different directions through the crystal. The relations 

 among the coefficients of elasticity, according to the system of 

 independent variables used in the preceding paper, which are 

 required to express such circumstances, may be investigated 

 by choosing for the normal cube a cube with faces perpendi- 

 cular to the lines joining the three pairs of opposite tetrahedral 

 corners of the dodecahedron. This choice of the normal cube 

 makes all the coefficients vanish except nine, and makes these 

 nine related one to another as follows : — 



W)« =wh "Wo ~ x+2 ^ 



(d\o\ /d?w\ /d?w\ 



and 



/ d 2 w \ / d 2 w \ / d?w \ _ 

 \dy dz/o \dz dx/ Q \dx dy/ 



where X, jjl, k are three independent coefficients, introduced 

 merely for the sake of comparison with M. Lame's notation. 

 In different natural crystals of the cubical system, such as 

 fluor-spar, garnet, &c, it is probable that the three coefficients 

 here left have different relations with one another. The body 

 would, as is known, be, in its elastic qualities, perfectly iso- 

 tropic if, and not so unless, the further relation 



/d 2 w\ __ / d 2 w \ (d?w\ 



\oWj Q ~ Wdz) + Z \~Wh 



were fulfilled. Hence the quantity k in the preceding formulae 

 expresses the crystalline quality which I suppose to exist in 

 the elasticity of a crystal of the cubic class. 



24. The fact of there being six axes of symmetry in cubic 

 crystals (diagonals of sides of the cube), has suggested to me a 

 system of independent variables, symmetrical with respect to 

 those axes, which I believe may be found extremely convenient 

 in the treatment of a mechanical theory of crystallography, and 

 which, so far as I know, has not hitherto been introduced for the 

 expression of a state of strain in an elastic solid. It is simply 

 the six edges of a tetrahedron enclosing always the same part of 

 the solid, a system of variables which might be used in alJ ex- 

 pressions connected with the theory of the elasticity of solids. 

 To apply it to express the elastic properties of a crystal of the 

 cubical class, let the tetrahedron be chosen with its edges parallel 



