( 132 ) 



there is a minimum value of q for the coexisting phases, viz. it 

 V2 = t\. Then the line joining these phases runs parallel to the 

 jc-axis, just as it runs parallel to the r-axis in the reciprocal case. 

 This too can only occur, if the coexisting phases lie on either side 



of the line — = 0; for the isobar that passes through the two points 

 dx 



of coexistence, can only have two equal values of v if between them 



maximum or minimum value for the volume occurs. The equality 



of v 1 and i\ for minimum value of q, to which we have concluded 



from the principle of reciprocity, follows from the simple equation, 



which holds for two successive points of a binodal curve, viz. : 



(v, — »,) dp = (« s — *•,) dq. 



For a pair of coexisting phases W^, is the same, and for a following 

 pair of such phases dM^ is also the same ; now the above equation 

 follows from dM l ^ 1 = v x dp — x x dq = i\dp — as t dq. If x, — x l = 

 aiK ] Vi — Vi is different from zero, then dp must be =0; in the 

 same way dq = involves the equality of w s and i\, if dp is not 

 equal to 0. We can also derive from this equation, how the nodal 

 lines lie on either side of the special nodal line for which a?, = a?, or 

 v. = v l , i.e. to which direction they diverge in a fanlike way. Let 

 us first consider the case x 3 = x 1} so maximum pressure on the 

 vapour-liquid binodal curve. On the left of this nodal line the sign 

 f v — Vi is positive on the vapour side, and the sign of dp, if we 

 do not limit ourselves to an infinitely small value of' dp, negative. 

 Then also the sign of (a? s — a;,) dq must be negative, and the sign 

 of dq being negative, a;, — .<■, must be positive. On the right of this 

 nodal line the sign of ?;, — i\ and of dp must be what it was in 

 the preceding case; but dq now being positive, ,i\ — ,i\ is negative. 

 So the nodal lines converge towards the vapour side. It would be 

 just the reverse if the pressure was minimum for x a = ai 1 , for then 

 dp is positive. Let us now consider the case v, = v u so minimum 

 value of q on the liquid-liquid binodal curve. Let us choose the 

 right side, so where a?„ > x x , and let us ascend, so put dp positive, 

 then q being minimum, dq will be positive. The second member is 

 positive, and so we find v, — 1\ positive, whereas for negative dp 

 the value of v a — v l would be negative. So the nodal lines converge 

 towards the right side, and we may consider the nodal line for 

 which v ' = «,, as axis of such a converging pencil. This shows us 

 at the same time how and where the plaitpoints must lie. As the 

 tangent to the binodal curve in the plaitpoint is to be considered 

 as the limiting direction of the nodal lines, both the p-line and the 



