Kinetic Theory of Solids. 529 



energy under a given stress. But if the separate components 

 u 2 , v'\ w 2 are not given, but only D the mean kinetic energy, 

 then, given P, Q, R, D, that is to say the stress and the 

 temperature as well as 5, H, Z, there are four relations 

 connecting the six quantities u 2 , v 2 , w 2 , f , rj, £. Two other 

 relations would make the connexion between energy and size 

 in different directions quite determinate ; it is the business of 

 experiment to ascertain these two relations. 



If the body is homogeneous and isotropic when free from 

 external stress, then the equations (2) are a little simplified 

 by the relation £ = H = Z = E, and we have 



&7?(?-e) 6|^ 



2r*W = 0, ... (3) 



with two similar equations. 



If, further, the body is subjected to hydrostatic pressure p 

 only, so that P=Q = R = p, then the three equations of form 

 (3) reduce to the one 



2D 1 



3?(^E)-^-6? S ^W = - • • • (4) 



Thus, if the size of the molecules is given, and also the law 

 of force, then the relation of pressure, volume, and kinetic 

 energy is determinate. Accordingly, we are able to calculate 

 the bulk-modulus of the solid (the reciprocal of its compres- 

 sibility) . 



Let co denote volume, then the bulk-modulus is coclp/dco, 

 denoted by 



But co=ve% where v is the number of molecules per unit 

 volume, and so 



* = I-s3s{?5=]S)-i Sr *w}- • • (5) 



We can go no further without the law of molecular force, 

 so I will now use the law of force, which I have formerly 

 discussed in connexion with gases and liquids (Phil. Mag. 

 5th ser. xxiv. & xxvii.), namely, <j>(r) = 3A??i 2 /r 4 . Then 



. ' 4:7rr 2 dr3Ampr 

 2,r<l>(r) — 



Phil. Mag. S. 5. Vol. 32. No. 199. Dec. 1891. 2 N 



