States, and the Thermodynamic State- Equation. 147 



This theory was at first regarded largely in the light of 

 •molecular theory *, to which it owes its origin. Its con- 

 sequences are, however, so important and so extended that 

 the empirical side of it soon came into the forefront, just as 

 van der Waals' equation for many purposes is treated as a 

 purely empirical thermodynamic state-equation. Moreover, 

 as other equations, such as those of D. Berthelot and 

 Olausius, were set up empirically to represent the facts more 

 nearly than van der Waals* equation, so it was realized by 

 many scientists, including van der Waals himself, that the 

 theory of corresponding states follows not only from that 

 equation, but from any state-equation with not more than 

 three specific constants. Meslin f first showed that this is a 

 purely mathematical consequence. For the general con- 

 ditions defining the critical point, namely (dp/'di')T : =0, 

 O 2 ;>/B ?,2 )t=0, when applied to the equation 



F(p,r,T, «,/3, 7 ) = 0, 

 .give p k , v ki T A . as functions of <x, /3, y : i. <?., «, 0, 7 in terms 

 °f Pk-> v k> ^~k> au d hence a new equation 



&(P, v, T, p k , v k , Tj = 

 is obtained, which must be of the form 



p v T 



\Pk v k U 



since p, v, T are measured in entirely independent units. 

 Furthermore, as Curie J also pointed out, any " critical 

 point " thus defined by general equations will serve the 

 same purpose ; and Berthelot § has, in fact, set up reduced 

 equations on the basis of three other critical points which have 

 certain special thermodynamic properties defined analytically. 

 For instance, experiment confirms the result deduced from 

 van der Waals' equation, that at each temperature there 

 exists one finite pressure (p) , at which the product pv has 

 the same value as it has at zero pressure and at that 

 temperature. Berthelot chooses as one of his critical points 

 the point, defined by the relation d(p) /dT = 0, at which (/?)■, 



* See, for instance, K. Onnes, Ak. Wetsch. Amsterdam (2) xvi, p. 24] 

 (1881) and Arch. Merland. xxx. p. 101 (1897). See also Happel, Pht/s. 

 Zeitschrift, vi. p. 389 (15)05), where an excellent resume' and bibliography 

 of the work on the theory of corresponding states is to be found. 



t C.R.cxxi(a), p. 136 (1893). 



t Arch. Sc. Phys. Geneve, xxvi. p. 13 (1893). 



§ Journ. de Phys. (4) ii. p. 186 (1903). 



L 2 



