112 Mr. H. Hilton on van der Waals' Equation. 



x Q 

 Each member has also an asymptote - — - = 1, which lies 



«/_ 



3 8 

 below the curve when x>j, crosses it when x — \, and lies 

 above the curve when x<~. 



The asymptotes all pass through the point (-|-, 0). 



A curve of the family (a) for which y is positive has a branch 



x 8 



running to one end of = 1, with an inflexion where £=1* 



1 y 7 



*& 



crosses the axis of x where x=^, and runs down to the 



■3> 



negative end of the axis of 6 as asymptote ; it has also a 

 branch for which x and 6 are always negative ; the curve for 

 which y = is similar, but has = as its asymptote ; a curve 

 for which y is negative has two branches as before, one for 

 positive and one for negative values of x, but in this case each 



branch cuts the axis of x in a real point I where # = + \ / — - 15 



these points lie further from or nearer to the origin than 

 x = -% as y J — 27 ; the curve for which y= — *27 touches the 



axis of x, where #=-3, and the corresponding branch lies 

 wholly below the axis of x (cf. fig. 3) ; the various curves of 

 the family lie so close to each other between x = ^ and x = 

 that they cannot well be distinguished in a diagram. The 

 parts of these curves lying between x = -| and x = 5— the parts 

 most interesting from a physical point of view — are given in 

 fig. 3, together with another curve discussed later ; fig. 2 

 gives a more complete tracing of two of the curves (y = 5 and 

 — 2*5) together with the curve (/3) (see below r ). 



Tables II. and III. give a series of values of 6 for different 

 values of y and x, from which the figures are drawn, putting 

 (a) in the form 



a-.y (vt v a\ 1 3(3#-l) 

 0- g (3*-l)+ Sv2 • 



The values of 3^ — 1 and v 2 — - are given in Table I. 

 The orthogonal trajectory of the family is 



4^(3^-1) ^j = -12^ 3 + 3(3^-l) 2 . 



