32 THE C DISCRIMINANT AS AN ENVELOPE. 



(b) represents two equal hyperbolae passing through the origin, with asymptotes 

 x= ±\=y. The curves (a) are a series of ellipses passing through the origin and 

 touching both parts of A = there, as well as at one other point on each. 



(3) c i x' i + 2c(y i -v i ) + (i/-xy(3f + 4:xy + 2x 2 ) = (a) 



A=(tf~-x*Y (b) 



y + x = is a locus of multiple points (c = 2x), y — x = is a particular case of the 

 primitive given by c = 0. 



(4) cXx* + f-l? + 6c 1/ (x* + f-l)-3(x>-2f-l) = (a) 



A=3(x*+if-lf = (b) 



The points (y = 0, x= ±1) are on all the curves (a) as well as on (b). 



All the curves touch at these points the discriminant which is a particular case of 



U = 0(c=+<x>); y = — is a tangent to all the curves at two points. The curve given 



by c = — a is the image in the x-axis of the curve given by c = + a, and as c increases 

 to + oo and decreases to — oo the curves and their images gradually close together, 

 and finally give the discriminant. 



