AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
255 
Art. 24.— If the parameters of both of the Surfaces having Conic Nodes become 
infinite, and E = 0 be the equation of the Envelope Locus, then A contains E 3 
as a factor. 
In this case it is necessary that both roots of the parametric quadratics ( 180 ) 
and (181) should become infinite. 
Hence the first differential coefficients of 
uv - W\ Nv - UW, U u - VW, 
with regard to any of the variables, must vanish on the envelope locus. 
[It may be noted that if 
_ W 3 ) = 0, and ~ {Nv - UW) = 0, then ~ (U u - VW) = 0. 
dx 
0 
^ (uv — W 2 ) = u x v + uv x — 2WW.,. = 0 
1 (V„ - UW) = V,® + v®, - uyw - uw, = o 
(182). 
Multiplying these equations by V, u respectively, and subtracting 
ufifv + TJfiNu - uvN x - W, (2WV - U u) = 0 ; 
therefore, using (158), after dividing by W, 
therefore 
u.TJ + U x u - WY, - W,Y = 0 ; 
i(U»-VW)=0.] 
Now A, dA/dx both vanish by (158). 
Next consider b 2 Afx 2 as given by (148). 
The first three determinants vanish by (158). 
The fourth and fifth determinants 
= 2 { v 4< wu - v ”>+ u 4( vw - u «)+- ws )} 
= 0 by the above conditions. 
The sixth determinant is 
2 
u x "W x N x 
W, v x U, 
V U w 
