THE DISCRIMINANT AS AN ENVELOPE. 31 



Practical Examination of the Curves U = and A = 0. 



Given therefore a system of curves U = and the discriminant curve A = 0, we may 

 draw the following conclusions : — 



(1) If the terms P and Q contain a common factor, (or if P = 0) this factor is 

 contained in the discriminant and is not an envelope but merely the particular case of 

 the C system obtained by putting c = 0. Similarly, if A and B contain a common factor, 

 (or if B~0), the corresponding discriminant factor is not an envelope. This factor 

 may be either single or repeated. 



(2) The remaining factors of the discriminant are either (a) single or (b) repeated. 



(a) Taking the single factors, we must test whether the corresponding values of c 



are constant or functions of x and y. Whether this is more easily done by 

 finding the value of A A /A B (as in the case where the c equation is of the 

 second degree in c) or by direct substitution, depends on special circum- 

 stances. If c be found to be constant the corresponding factor is not 

 envelope ; but if c be variable, the curve is a true envelope and is touched 

 at every point of its length by the C curves. 



(b) Taking the repeated factors we have to test whether they represent doubled (or 



in general n-pled) curves, at every point of which U* = U v = in virtue 

 of U = or loci of multiple points. * 



The value of c discriminates between these two cases. If it be constant, the 

 locus is a particular case of the C curves. An example of this is supplied 

 by the system 



C \x 2 + y l _ 1)2 + g^ + y 2 _ !) _ 3(x 2 _ 2 f _ 1) = 



which is discussed below. 



If c be variable, the locus is a locus of multiple points, and in general is nothing 

 more. 



The following systems of curves illustrate some interesting points. 



(1) c%x 2 + y 2 + 2x) + 2cy + 2x=0 (a) 



A=y' 2 -2x(x 2 +y 2 + 2x) (b) 



A = has a node at the origin and an asymptote 



x 



Every circle of the system (a) touches A = and passes through the origin, and 

 two circles of the system (given by c = ±1) touch A = at (0. 0) 



(2) c 2 x 2 + c(x + y) + y 2 = (a) 



/\=4:xY-(x + y) 2 = (b) 



* Note also here that the vanishing of the first three or the last three coefficients, in virtue of their containing 

 a common factor, leads to a repeated factor in the discriminant. This case has already been disposed of under (1). 

 VOL. XXXIX. PART I. (NO. 4.) G 



