824 ^-DISCRIMINANT OP A DIFFERENTIAL EQUATION OF THE FIRST ORDER. 



(1, 0, 0), once at (1, -1,0), and twice at (1, 1, c). The family has no envelope ; but 

 merely the node locus x — y = Q. 

 The c-discriminant locus is 



TIlC cuotc 



9c*x 2 - 4cx - (12if + \2y 2 + 4y) = , 



or 



x-^{l±(3y+lf} 



has a fixed inflexion and inflexional tangent at oo and passes through the fixed points 



A. 



(0,0), 0, 



-S + ij3 



"), (o, 



-3-ij3 



■)■ 



It has a cusp at (2/9c, - 1/3). The intersections of two consecutive curves are (oo , 0) 

 thrice, A, B, C and the cusp thrice. There is therefore merely a cusp locus (fig. 4). 



Fig. 5. 



The c-discriminant locus is 



16x%Sy+ 1)3 = 0. 



x 2 = corresponds to a degenerate curve of the family for which c=co . 



The simplest case of all in which the envelope degenerates into a point-discrete is 



the linear pencil 



cu + v = , 



where u and v are integral functions of x and y. In this case each curve intersects 

 every other consecutive or non-consecutive in the points of intersection of u = 0, v = ; 

 and the corresponding differential equation of the first order and first degree has, as is 

 well known, no envelope singular solution. The proper consideration of this case alone 

 is sufficient to show that no theory of envelopes which takes no account of the manner 

 in which the arbitrary constant is involved can be of much value for the theory of 

 differential equations. 



