AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
259 
(B.) The Envelope Locus is z = 0. 
The tangent plane at £ g, £ is 
(X - f) 2a (of + h + g) + (Y — g) 2 (of + 6 + if) + (Z - Q ( 2 « 8 £ + l) = 0. 
Hence at the point f, 17, £, where 
+ & + t? = o» c — 0 , 
the tangent plane is Z = 0. 
Hence the factor z 3 is accounted for. 
(C.) The Parameters of both Surfaces having Conic Nodes are infinite. 
In this case 
u = z 2 -\~ x 2 , v = 1, iv = y 2 z, U = y, V = xy, W = x. 
H ence the equations 
a 2 | ( MW _ W 2 ) + 2a ~ (Vv - UW) + ^ (iw - U 2 ) = 0, 
and 
6 2 1 ( uv _ w 2 ) + 26 (U« - VW) + | («w - V 2 ) = 0, 
become, when 2=0, 
(0) a~ -J~ 0 (a) -f- 1 0, 
(0) 6 2 + 0 (6) + tf 2 = 0. 
Hence both roots are infinite. 
If the differential coefficients in the parametric quadratics had been taken with 
regard to x or y, the equations would have been wholly indeterminate. 
Art. 25 .—If the parameters of both of the Surfaces having Conic Nodes become 
indeterminate, then at every point of the Locus of Ultimate Intersections there are 
an infinite number of Biplanar Nodes; each Surface of the system has a Binodal 
Line lying on the Locus oj Ultimate Intersections, and if the locus of these 
Binodal Lines be B = 0, then A contains B 4 as a, factor. 
In order that the parametric quadratics may become wholly indeterminate, the 
first differential coefficients, with regard to each of the three variables, of uv — W 2 , 
U u — YW, Yv — UW, vw — U 2 , uw — V 2 must vanish. These involve the 
vanishing of the first differential coefficients of UV — Wiv. 
It will be shown, first of all, that the ratios ( 164 ) are in this case equivalent only 
2 1. 2 
