( 189 ) 



d'p d 3 pdv d*p 



But then not only = in the equation -7-^, — + - — — = 0, but 



dv dx dv* dx dvdx 



d'p dv 



also — . The value of — is then indefinite, and at the temperature 

 dv* dx 



at which this takes place, and which is the minimum temperature 



8 a x dp 



represented by — — , the curve — has two branches, which intersect 

 27 b x dv 



1 db 1 da 



in the point given by v = 36 and x, belonging to = — — . For 



b dx a dx 



higher values of v, e.g. v = 4è, the point of intersection of the two 



dp 

 curves lies on the vapour branch of — = 0, and vice versa. If we 



dv 



1 da db da 



. 2 a dx , dx dx 



write v = nb, the form - = - — — follows from - — — . For 



n — 1 1 db v — b 2a 



b dx 



those values of x for which the numerator is smaller than the deno- 



da 

 minator n ]> 3, and vice versa. Only if also — = should occur in 



dx 



the diagram, the value of n, and so also of v, is infinite. 



If we determine the point in which the two curves touch, we 



d*p 



shall find the same point in which has the minimum volume ; 



dv dx 



dp 

 for as the curve — = has a tangent parallel to the Jf-axis in 



dv 



d'p 



every point of intersection, also the curve = must have such 



dv dx 



a tangent in case of contact. 



dv 

 The condition that for a point of the last-mentioned curve — = 



dx 

 d'p 



is = 0. So we have 



dv dx 2 



and 



dv dx (v — by 



MRT ( — ) — 



d l p \dxj dx* 



— — = or 3 ±-^- — — 



dvdx' (v — b)* v* 



13* 



