States, and the Thermodynamic State- Equation, 149 



At the same time there is indicated an extension of the 

 theory to phenomena involving other than the purely 

 thermodynamic quantities p, v, T. 



One point revealed by the work of Meslin, Curie, Berthelot, 

 and others has perhaps not been sufficiently emphasized, 

 namely, the fact that the theory of corresponding states is in 

 its foundations quite independent of any " critical point/' 

 The very essence of its truth is contained herein : by as- 

 signing to each substance special or specific units of p, v, T, 

 three specific constants are eliminated from the state-equation. 

 The usual method of assigning specific units is to put 

 p,=v„=T, = l for each substance, where k refers to some 



arbitrarily selected critical point. 



There is, however, a directly mathematical method of 

 ■choosing specific units which is more fundamental and 

 more general. Consider the general case : — The variables 



.1?, y, z, , involving n independent dimensions, are related 



to one another by the equation : 



F(#,#, ~, , «, & y, ) = 0, 



•containing r specific constants a, /5, 7, Substitute for 



these constants certain functions of them, n of which 

 {a, b, c, ) are of the dimensions of x, //, z, re- 

 spectively, and the remainder (X lf X 2 , ) of zero dimension. 



The equation must then be obtained in the form 



G 



*&f»; > Xis X2 ' ) =0 - 



^l a — ^-> y\b = V, z l c = & & c -^ may now be called the reduced 

 variables, and the reduced equation is written 



G&17, ?, ,X!,X 2 , ) = o. 



In the case when n<£r, there are no constants (X) of zero 

 dimension. The reduced equation 



Q(tv,& )=o 



is then identical for all the systems to which the original 

 equation applied. The purely algebraic process reduces the 

 equation to a non-dimensional form, containing n specific 

 constants less than the original one. 



To apply this method to van der Waals' equation (1) 

 above. Here a — ^^ 2 ^ b = fi, r = Ma/R/8 are the constants 

 of dimensions of />, r, T respectively, and the " algebraic " 

 reduced equation is : 



(»+y)'(*-' 1 >-« w 



