548 Mr. W. Sutherland on a 



Tomlinson (Phil. Trans. 1883) has investigated the same re- 

 lation. We have seen, in the experimental introduction, that 

 the elastic properties of the metals correspond only at tempe- 

 ratures which are the same fraction of the melting-tempera- 

 tures, and accordingly any relation connecting Young's 

 modulus at a fixed temperature with molecular volume or 

 domain must be empirical. Now as the coefficient of expan- 

 sion b is roughly proportional to the molecular domain m//?, 

 the relation Qbm/p = constant becomes Q(m/p) 2 = constant ap- 

 proximately, whence we see the origin of Wertheim's relation. 

 There are various other approximate empirical relations dis- 

 covered by different physicists which correspond to combina- 

 tions of those already established, on theoretical grounds in 

 this paper, but there is no use iu discussing them further. 



8. Ratio of Lateral Contraction to Elongation in Young's 

 Experiment. — We can now take up the famous question as to 

 whether the ratio of lateral contraction to longitudinal ex- 

 tension in the Young's modulus experiment has the same 

 value for all isotropic bodies, as asserted by various builders 

 of statical molecular theories of elasticity, and as denied by 

 the upholders of a science of elasticity apart from molecular 

 considerations. As no statical theory founded on central 

 forces can give a true account of rigidity, it seems useless to 

 discuss a deduction of such theories, but the important point 

 of getting a test for isotropy is involved. We see at once, 

 from the results of the experimental introduction, that tem- 

 perature is an important condition which has never been 

 taken into account by the upholders of a constant ratio. In 

 isotropic solids we have the following equations for a the 

 ratio of contraction to elongation : — 



(T=(U-2n)/2(U + n) and cr = q/2n-l. 



Now we have seen experimentally that 



n=N{l-(*/T)*}, 



and by theory (11) 



, 2 Jcm(l-28bd) 



f mWm/p ' 



so that a is a not simple function of the temperature. At 

 absolute zero k is according to our theory very large, and 

 may be regarded as infinite, while n remains finite, and there- 

 fore the value of a at absolute zero is \. But again at 

 the melting-point n is zero, while k is finite, so that at the 

 melting-point cr again attains the value f . Now a has been 



