544 Mr. W. Sutherland on a 



shown in the compilation of data given in the introduction. 

 These disagreements would seem to indicate that even if these 

 metals can be got in an approximately isotropic state at all, 

 their isotropy is easily disturbed, but this point will be re- 

 turned to immediately. The mean value for the ratio Q/N 

 for the 7 metals Cu, Ag, Au, Mg, Al, Fe, and Ni is 3*0, 

 and accordingly it appears that these metals may be regarded 

 as isotropic, and as having molecules so nearly incompressible 

 as to give the relation Q=3N characteristic of incompressible 

 "molecules. 



The study of the ratio Q/N leads naturally to the famous 

 controversy amongst elasticians as to whether there is a fixed 

 value of the ratio of lateral contraction to longitudinal exten- 

 sion for all isotropic bodies subject to traction ; but this 

 matter had better be postponed a little until we have seen 

 what account the kinetic theory can give of rigidity. This 

 is the most important point in a theory of solids, and as I 

 have already pointed out that no static theory can give an 

 adequate explanation of rigidity, there is great interest in 

 seeing how the kinetic theory will fare. 



7. Rigidity according to the Kinetic Theory. — The funda- 

 mental equation for a solid, homogeneous and isotropic and 

 free from external force is 



2D 1 



09 &»(«-E)~6? Sr *to =0 ' 



or 



3?^E)-V=0. 



Suppose the solid now subjected to a pure shearing stress 

 specified by a traction P parallel to the axis of x, and a pres- 

 sure P parallel to that of y : the traction counts as a pressure 

 — P, and the corresponding strains are an elongation of e to 

 £• parallel to #, and a contraction of e to rj parallel to y. The 

 strained body is no longer isotropic, and equations (3) are 

 applicable to it if we write — P for P, and put Q = P and 

 R = 0. Let £ = e + S£, and r) = e + Srj, and f= e, then $r]= — Sf. 

 The effective kinetic energies in different directions are no 

 longer equal each to D, but become \mu\ ^mv 2 , and |mi« 2 , 

 which will be denoted by D 1? D 2 , and D 3 , and let D 1 = D + SD 1 

 and so on. We do not know at present how we are to ex- 

 press the condition that the shear is made at constant tem- 

 perature, for when the effective kinetic energy is different in 

 different directions, w T hat is the relation between these dif- 

 ferent energies and temperature ? All that we can assert is 

 SD 2 =— SDp Now the shear produces a change of volume 



