( 199 ) 



1 



— 0,15 + 0,25* -J 0,25 x* = 



4 



or a; l nearly equal to 0,5. 



Here we very clearly get back the first case. 



The intermediate case would require that a?, should be equal to 



t ï, . If we wish to direct our attention to other particularities of the 



intermediate case, we observe: 1. that then there is only one point 



dp d l p 



of intersection for — = and — — = at every temperature ; 

 dv dvdot 



dp 

 2. that then at the double point of — = one of the branches must 



dv 



have a tangent parallel to the X-axis ; and so, the two values of 



dv 



— for that double point being given by the equation : 



dx 



d 'P( dv \ | o d% P ( 

 dv i \dxj dv*dx\ 



dv\ d*p 

 + 



dv*dx \dxj dvdx* 



d s p d % p 



- is again equal to zero (see page 195). The curve — — =0 



dvdx* dvdx 



now does not pass through the double point either with its descending, 



nor with its ascending branch, but has there minimum volume. At 



dx) 

 lower temperature the vapour branch of — = is cut in a point with 



dv 



somewhat lower value of ac, and at higher temperature the liquid 



branch of the righthand branch is intersected with slightly higher 



d'p 

 value of x. Just at the temperature of the double point — — = 



dvdx 



touches with a tangent parallel to the X-axis. 



If more in general, we wish to determine what the ratio of 



da 



( 



dx j a x 



m is at that value of x for which — has minimum value, we 

 a* a b x 



a — 

 dx' 



may take the following course. From : 



da db 



b — = a — 



dx dx 



we derive 



1 1 + (n-1) x\ [B + Ol = ^-i (A + 2B,r + Cx'-) 



-j 



or 



