580 Mr. H. Hilton on van der WaaW Equation. 



x = and y = at infinity, so that x and y for the branch are 

 always negative. 



If #=0, the curve breaks up into the straight line ^ = | and 

 the curve yx 2, + 3 = 0; these two branches meet at the point 



<!-, -27). 



If is negative, there is a branch between x = co and 

 x=\ touching y = and x=-^ at infinity, y for this branch 

 being always negative; a branch between x = ^ and x=0 

 touching these lines at infinity, so that near x=\ the ordinate 

 of the curve is positive and near x = negative (the branch 

 crosses the axis of x and has an inflexion, but cannot in a 

 diagram be well distinguished from a straight line) ; and a 

 branch on the negative side of x = for which y is negative 

 near x = and positive near x=— go (the branch crosses the 

 axis of x, has a tangent parallel to this axis, and an inflexion). 



Any member of the family can be readily traced by taking 

 a series of values for x and calculating the corresponding 

 values of y. Fig. 1 shows as much of the curves 6 = 2 an( l 

 0= — 1 as can be conveniently put on a diagram. Fig. 2 

 shows the part of the curves lying between x=\ and <2?=5 

 (which is the part interesting from a physical point of view) 

 for 19 positive values of 6. 



The tables given below show the values of y calculated to 

 three places of decimals from which the figures were drawn. 



Any member of the family is of degree 4, of class 5, and 



of deficiency zero ; it has two double points and one cusp 



(coinciding in the multiple point at infinity) ; two bitangents; 



and four inflexions (of which two are always imaginary, and 



91 X7 \ 

 the other two are also imaginary if 6 > n ^ \ 



The area included between any curve of the family and 

 the lines y=0, x x =0, x 2 =0, is 



80 3^ 2 -l 3^ J3 

 3 ® 3.%— 1 x 2 x{ 

 The orthogonal trajectory of the system is 



x\Zx- 1) = d £ (3afy-9# + 6). 



The radius of curvature at any point is 



^ 6 (3.a?~l) 2 + ( 24ay 3 - 6 3^- 1 2 ) 2 } * 

 18^ 5 (3^-l) 3 [8^ 4 -(3^-l) 3 ] 5 

 if the tangent at the point is parallel to y = 0, this simplifies into 



3 4 (3ff-l) 



18 (!-«*•) * 



