L 579 J 



LVI. A Note on van der WaaW Equation, 

 By Harold Hilton*. 



BEARING in mind the great importance of van der Waals' 

 equation, it seems worth while to collect the various 

 mathematical properties of the family of curves which is the 

 graphical representation of the equation, together with some 

 accurate tracings of several members of the family. 



The form of van der Waals* equation we shall consider is 



80 3 /. 



^=3^1"^ (a) 



or, as it may be written, 



3#a? 3 -(y+80)# 2 + 9#-3-O, 



where x, y, and 6 are respectively the " reduced " volume, 

 pressure, and temperature; i. e., the volume, pressure, and 

 temperature expressed with the critical volume, pressure, 

 and temperature respectively as units. 



The equation [a) may be considered as representing a family 

 of curves of the fourth degree, of which 8 is the parameter. 



The curve (a) cuts the axis of x in finite points, where 

 80A' 2 — 9<#-f 3 = ; the roots of this equation are unreal if 

 0>§; both equal to | if = %; both positive and >~, if lies 

 between f 2 and 0; one root is infinite if = ; both roots are 

 real, one lying between — oo and 0, and the other between 

 and -^, if is negative. The curve cuts the axis of y in no 

 finite point. The curve has the axis of x as an ordinary 

 asymptote, and a triple point at infinity in the direction of 

 the axis of y, at which ^=3 is the tangent to one branch, and 

 the tangents to the other two branches coincide in x=0. 



Hence to every value of x there can be at most one finite 

 value of y ; and to every value of y there can be at most 

 three finite values of x of which two may be imaginary. 



If is positive, the curve has a branch between x = co and 

 # = 3, touching y = and x = \ at infinity; so that y is in 

 each case positive; a branch lying between x=\ and x = r 

 and touching these lines at infinity, so that y is in each case 

 negative (in fact for most positive values of Q this branch lies 

 altogether so far on the negative side of y = that it cannot 

 well be drawn on any diagram, and hence does not appear in 

 fig. 1) ; and & branch on the negative side of x = touching 



* Communicated by the Author. 



