AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 155 
Hence the locus of ultimate intersections is 
[</> y> z)J = o. 
(B). The Envelope Locus such that two consecutive Characteristics coincide is 
</> (x, y , z ) = °- 
Let £ y, C be any point on if) (x , y, z ) = 0. 
Take a = (£, y, £), and consider the single'surface 
<f> ( x , y, z) + l> (x, y,z) — xfj (£ y, £)] 3 = o. 
Put X = £ + X, y = 7) + Y, 2 = i + Z : then the lowest terms in X, Y, Z are 
i 1+’ i !+ z S 
Hence the tangent plane to the surface at g, tj, £ is also the tangent plane to 
{x, y, z) — 0 . 
Hence </> (x, y, z) — 0 is the envelope, and the equations of the line of contact are 
<f) (x , y, z) = 0, xj/ (x, y,z) = \ji (£, y, £). 
Now the equations corresponding tof=0, Df/Da = 0, D^Da 3 = 0 are 
<£ (x, y, z) + [i ji (x, y, z) — of = 0, 
- 3 [xjt (x, y, z) — af = 0, 
6 [xfj ( x , y , z) — a] = 0. 
These are all satisfied by the coordinates of any point on the line of contact. 
Hence two consecutive characteristics coincide. This accounts for the factor 
[j> ( x , y, z)] 2 in the discriminant. 
Art. 8.— Consideration of Loci of Binodal Lines, which are also Envelopes, 
(A.) It wall be shown that this is the case reserved in Art. 1, viz., where 
Ds/i/Dcq 2 = 0. 
The equation of the biplanes is 
K f] (X - f f + b, r,-] (Y - v f + [£, £] (Z - if 
+ 2b.a(Y-,)(Z-£) + 2[£,f](Z-£)(X-f) + 2[f,,](X-f)(Y-,)^0 (53). 
This breaks up by (29) into the two planes 
x 2 
