Density, Temperature, and Pressure of Substances. 341 



where R a is the same for corresponding states. This follows 

 since the above two equations may be written 



n,ip c -=axp CJ and ?i 3 T c = &/3T c , 

 and therefore 



a=— , and o=- 3f 

 a p 



or J and a have the same values at corresponding tempera- 

 tures. Thus the equation in question is proved for all states 



of matter. 



1"RT 

 From this and the above equation we obtain pv = , 



which applies to all states of matter, where I is the same at 

 corresponding states. 



The general equation of state can be deduced from one of 

 the foregoing equations, but it is of a form which is not 

 of any use. We have obtained 



where 



<m i 4 h 



\dv »V8-"3 „7/3J 



and is therefore a function of v and T. This is a linear 

 partial differential equation, and the Lagrangian subsidiary 

 equations are therefore 



dv dT dp 



T -p + k 



An independent integral of these equations is m =A, and 



let the other be denoted by ^r f) (T, v, p) = B. Then an integral 

 of the partial differential equation is given by 



fag,^(T,r, ? ))=0, 



where ^ 6 is an arbitrary function. One of the conditions 

 which determines the form of the arbitrary function is that the 



RT 



equation must approximate to pv — when the matter is 



in the gaseous state and the pressure is lowered. 

 London, December 19, 1910. 



