( 625 ) 



v da 



more and more, and as — can never become = J, because 



b dx 



cannol become infinite, the curve v = b is the second asymptote. 



So if ,v is made to increase more and more, also beyond the values 



which for a given pair of components are possible in order to 



examine the circumstances which may occur with all possible systems 



db 



for which with positive value of — increasing value ot Tk is always 



dx 



fdp\ 



tound, a minimum volume must occur on the curve - =0. So 



\dxJ vT 



/d*p\ 



for this point —— = 0. 



Now that we have described in general outlines the two curves 

 which control the course of the //-lines, we shall have to show in 

 what way they do so. 



From 



(-) 



dv\ \uxJvT 



dxJpT (dp 



\dv JxT 



follows that to a p-line a tangent may be drawn // ,/-axis when it 



f dp \ 

 passes through the curve I — I , and a tangent // r-axis, when it 



fdp\ 

 passes through the curve — . But though these are important 



\dvJ.rT 



properties they would be inadequate for a determination of the course 

 of the isobars, if not in general outlines the shape of one of these 



fdp\ 



lines could be given. The line - = viz. intersects the line 



\dxJ vT 



j — J = in two points, and it is these two points which are of 



fundamental significance for the course of the y;-lines. The point 

 of intersection with the liquid branch is viz. for a definite //-line a 

 double point, the second point of intersection being such an isolated 

 point that it may be considered as a p-curve that has contracted to 

 a single point. The surface p=f(,v,v) is namely convex-concave 

 in the neighhourhood of the first mentioned point. Seen from below 

 a section // r-axis is convex, and a section // .i'-axis is concave. A 

 plane, parallel to the v, «-plane touching the //-surface intersects, 

 therefore, this surface in two real lines, according to which /> has 



