( 328 ) 



7. Another form for the equation of* the ellipse. 

 Most of the above results may be obtained by considering that the 

 equation of the ellipse may also be written in the form: 



i it/-' r i («— i) 2 ) 



(l + 8i ) + n 2 (l + f 3 )-^— ^(n-l)M =4n»Hl+v- ^ J 



/r 1 • / 2 



The (irsl member pul equal to O is the equation of the line A'B\ 



and the factors of the second member, put equal to are the equations 



of tangents lo the ellipse in the points B' and A'. 



If' the ellipse intersects the (irsl parabola, and so also twice the 8, 



axis, and if therefore pari of the ellipse lies within the (irsl parabola, 



then there is a continual series of points which yield sets of values 



of 8j and (■,, for which no three-phase-pressure is to be expected. 



This series of points begins where the ellipse cuts the first parabola 



in the J M or lowest point, and terminates either in the second 



point of intersection or in the 5, axis. In the latter case when the 



second point of intersection with the g, axis lies higher than (n — J) 2 . 



8. Ratio of the critical temperatures of the components. 



a, 1 -(- f-, "., /r(l 4 f 2 ) 



For a point of the ellipse is = - and = - .and so 



1 l c (>— l) s c (n- 1)- 



- = II' from the point 0' (e , = — J and f, = — J) we 



draw a line to such a point, and if we put the angle which this 



line makes witli the r-., axis equal to </, then - - = tg<p, and so 



l + «. 



also - = w cot y. or tg(p=.n— . 11' we put the question whether all 



the points of the ellipse can occur, we notice first of all that the 

 condition a l3 ]> already excludes the points lying below the line 

 P' Q. Let the ellipse be inclosed in a rectangle, the sides of which 

 are parallel to the axes, let us draw the diagonal through the centre 

 for which to <f = ir from the point , (e 1 = — 1 and e 2 = - 1). 



r U: 2 1 



This diagonal cuts the ellipse in a point that belongs to — — = — . 



For all the points of the ellipse lying right of this point, tg<p^>n', 



T h , 1 " T k . I' 



and so —<d ■ Thus we should have — = — for the highest point 



T ki n Ijcy n 



of' the ellipse. We cannot assert with any certainty that such cases 

 do not occur. Substances with larger molecules have not a necessarily 

 higher T k , and that the critical temperature can be lower with ■//""> 1 



