( 195 ) 



dv 

 branch has two coinciding values óf x, for which = 0, and so 



6 da 



also a point of inflection, namely for the volume that is the smallest 



volume for which a point of intersection of the two curves exists. 



At still lower temperature the liquid branch has no longer a point 



of intersection ; but the point of intersection of the vapour branch 



continues to exist, and proceeds continually to smaller value of x. 



I need hardly point out that in this description negative values of*' 



are again not considered as unreal. The condition for a point occur- 



dp dv d" v 



ring on the curve — = in which — = and — = is found from : 



do dx dx* 



d*p dv d*p 



— 1 —-0 



dv* dx dvdx 



and 



d*pd'v d'p /dv\* d l p fdv\ d*p 



do* dx* 



v d*p fdv\* d s p /dv\ d*p 



- + — ( — I + 2 — =- ( — | H — = 0. 



1* dv 9 \dvj dvdx \dxj dvdx* 



dp d*p d s p 



Hence besides — = 0, also = and = 0. 



dv dvdx dvdx* 



Let us now consider the second case, for which the value of x 



/day 2 d-a 

 corresponding to — )= — a — , is the smallest. For this value of 

 \dxj 3 dx* 



x the value of T is then maximum, and the temperature for the 



dp 

 double point of — = will be a minimum. This means that with 

 dv 



decrease of T two points of intersection vanish, whereas in the 



preceding case two new points of intersection appear with decrease 



of T. 



Let us first consider this minimum temperature; then a curve 



d*p 



■ = passes through the double point, which, in this point, may 



dvdx 



(ID 



be considered to have two points in common with the line — = 0, 



dv 



and which has a third point of intersection for smaller value of x. 



This third point of intersection is to be found on the vapour branch 



dp 

 of the lefthand branch of — = 0, because it has smaller x. Fig. 35 



do 



indicates the places of the three points of intersection for this value 



of T. With decrease of T two of the points of intersection lie on 



dp 

 the vapour branch of the lefthand branch of the line — =0, and a 



dv 



