Laws of Molecular Force. 17 



are in contact, and $>{r) the attraction between two molecules 

 at distance r apart, the summation to extend to all molecules 

 within appreciable action of any one. It may be more con- 

 venient to sum both members of the above equation for all 

 the molecules in unit mass, when we get 



- 1 * 22r£(r) = (4) 



a#(e-E) Qe 



If the law of force is given by <j>(r) = 3 Am 2 /?' 4 , then XX?'$(r)/6 

 reduces to what is denoted by (p in the notation of the " Laws 

 of Molecular Force/' and e 3 = m/p, so that 



22D ■ 2/ , M2/ /M\ 2 22mD fK { 



This equation was established on the assumption that the 

 molecules collide as though they were perfectly restitutional 

 spheres. 



Now 22 D is tw T ice the kinetic energy of the motion of the 

 centres of mass of the molecules in unit mass, and, if the 

 energy of other motions is negligible, equals 2Jc9, where is 

 the temperature and c the specific heat. e 2 (e — E) = E 3 (<?/E — 1) 

 approximately; if the molecules are invariable with tempera- 

 ture, *»/E — l = bd, where b is the coefficient of linear expansion 

 of the solid, and it was shown that the metals behave as if E 

 diminished with rising temperature in such a way as to make 

 ^/E — l = 7b6 approximately ; and as W=m/p nearly, we have 



m22D 2mJcO , '^ au/ljr/ . ta 



3?(^E) = 2l6^ =2JcM / 21 ^ M /' ,) ' • • (6) 



m being the actual mass of a molecule, and M the ordinary 

 molecular mass referred to that of the hydrogen atom. Now 

 cM, by Dulong and Petit's law, is nearly 6 "4 for the metals, 



sotnat Wl='6U{M/p)/b (7) 



The values of b have not been found experimentally for a 

 number of the most interesting metals, but can be got by 

 means of an empirical relation given in" A New Periodic 

 Property of the Elements " (Phil. Mag. ser. 5, xxx., also 

 xxxii. p. 540), namely, if T is the absolute melting-point 

 bTm = '0U, and then 



M 8 /=-61J(M//d)TMS/-044. ... . (8) 



Taking J as 4*2 x 10 7 ergs and using 10 12 dynes as the unit of 

 force as hitherto, then finally 



M 2 / = 5-8xlO- 4 (M/p)TM£ (9) 



This equation applies, so far as we know at present, only to 

 the metals ; it has been deduced from (3), which applies only 

 Phil. Mag. S. 5. Vol. 39. No. 236. Jan. 1895. C 



