150 Mr. Gr. von Kaufmann on the Theory of Corresponding 



Since p k =-^ja, r,=36, T^-fac, the "critical" reduced 

 equation (2) is obtained at once from this by putting 

 7r=p/a=^j7r', &c, &c. Similarly, Berthelot's reduced 

 equation (3) and any other " critical " reduced equations 

 follow from (4), which is the fundamental and the simplest 

 form. 



The equation of Clausius : 



[*+i^]<-«-£- T • • • (5) 



gives the algebraic reduced equation 



b + ^f+xrlu-v= d < 6 > 



Here 



a=(«K/£ 8 M)», b = j3, c=(«M//3R)*, X=y/£. 



Clausius' equation contains four constants. Hence there 

 appears, in the reduced equation, one constant X of zero 

 dimension. By putting A,= in this equation, the reduced 

 equation derived from Berthelot's state-equation : 



(p + u/Tv*)(v-l3) = KT/M., 

 is given, 



This purely algebraic method of obtaining the reduced 

 equation is thus essentially similar to the old method, and 

 directly expresses the theory of corresponding states. The 

 algebraic reduced equation involves the reduced equations 

 obtained by the use of any critical points, in all their 

 consequences : it is the simplest form and the most general. 



The great use of this method is that it opens the way 

 to applying the theory of corresponding states to other 

 phenomena which do not involve easily recognizable critical 

 points. The physical facts corresponding to the above 

 mathematical process are, in general, these : — The variables 

 represent a certain set of energies and their factors (and, in 

 certain cases, time), the relations of which are exhibited by 

 a variety of material systems. Underlying the relations of 

 these quantities are certain natural laws, within whose scope 

 those relations may differ in the different systems. Now a 

 certain degree of this difference is always expressible by 

 choosing the units of the quantities specifically in the different 

 systems. And if the degree of difference be not too great, 

 the whole of it is thus expressible. In that case, the sets of 



corresponding quantities (a/a, y/b, ) are identical in all 



the systems; just as the sets of quantities (a, y, ) would 



be identical if the natural laws gave no scope of difference 



