1887.] for the Change of State from Liquid to Gas. 221 



by the usual methods, owing to the instability of the states they 

 represent. 



Clausius's equation may be put into the following form by assuming* 

 p '(RT) = y and v — cc — x — 



yix+^y = 0r + 7 ) 2 - § • 



From this it is evident that an isothermal is a quartic curve having 

 asymptotes y — 0, x = 0, and aj + 7 a double asymptote at a cusp at 

 infinity, so that the point at infinity on this line is a multiple point of 

 a high order. 



If we calculate the positions of the points of tangency of tangents 

 parallel to y = 0, for which consequently dyjdx — 0, we have the cubic 

 equation — 



and when two of its roots are equal Q c = and this determines the 



critical isothermal. The quartic consists of three branches. One, a 

 serpentine branch, lies in the positive region of x and mostly of y, and 

 is the only branch of physical interest at present. The other two 

 branches lie entirely in the negative region of x and y. One of these 

 is somewhat parabolic, and lies between x = and x — —7 asymptotic 

 to both of them, the other is hyperbolic and asymptotic to x = — 7 

 and to y = 0. The always real solution of the cubic that determines 

 the points where cly/dx = for positive values of and 7 is a point on 

 the parabolic branch of the curve that lies between x — and x — — 7. 

 The other two roots, when real, determine the highest and lowest 

 points on the serpentine part of the curve that lies in the positive 

 region of x. The accompanying diagram represents the general 



Fi&. 1 



