110 Mr. H. Hilton on van der Waals Equation. 

 The orthogoDal trajectory of the family is 



The integral of this is the result of eliminating p between 



27 



y+ p (l-Sp) 2 

 and 



g - 13 , r is X3 + *p> _ up ■**-'£& + _^_\ 



\845(l-8p)« 4225^/65(1+^) Vl+j^J_" 



Any member of the family may be put in the form 



L_-!=i. 



3(aj?-i) 3 



8# 2 «■ 



The values of the denominators for different values of x are 

 given in Table 1. ; it will be noticed that the point where the 

 line cuts the axis of y travels from zero to -co as x goes 

 from co to 0, and from — cc to zero as x goes from 3 to — qo ; 

 while the point where the line cuts the axis of 6 travels 



from zero to § as x goes from go to |, 



from § to — go as x goes from § to 0, 



and from — go to zero as x goes from to — go . 



When # = |the line is = 0, and when # = c© the line is 

 y=0. 



In fi.g. 1 the lines of the family are drawn for various values 

 of x (in some cases for the sake of clearness only the extre- 

 mities are shown) , together with the envelope and another 

 curve discussed later. 



Again, let us take the equation 



3yar 8 -(y + 80)a 8 + 9a?-3 = 0,.. ... (a) 



iind consider y as a variable parameter. 



The equation now represents a family of curves each of which 



is of the 3rd degree and class, has one cusp and inflexion, no 



double points or bitangents, and is of deficiency zero. The 



cusp is in each case at infinity, the tangent being ^ = ; and 



3 + y 

 the inflexion being where a=l 9 = -, the tangent at the 



inflexion being 



(•-*?)- §&-D(.-i>. 



