AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 163 
The first of these touches the locus of binodal lines. Hence the locus of binodal 
lines is also an envelope. 
Hence the factor [</> (x, y, z)] 3 of the discriminant is accounted for. 
(C.) The Surface </> (x, y, z) + -f- = 0 is an Envelope. 
For seeking its intersection with 
{x, V, + (p {x, y, z,) [iff (x, y, z) — a] + [v/i (x, y, z) — of = 0 , 
it follows that 
bp {x, V, z) - aj — -TT ( X, y,z ) — a] + ffy = 0 . 
Put 
therefore 
therefore 
xfj (x, y, z) — a — 77/9, 
y 3 — \2r) + 16 = 0 , 
(y — 2 Y (y + 4 ) = 0 , 
i.e., (9 [xb (x, y, z) - a] — 2 } 3 (9 [xjj (x, y, z) — a] + 4} = 0. 
Consider now any point g, 77, £ on the surface 
for which 
and 
[<j> (x, y, z)J + cj) (x, y, z) [xjj (x, y, z) — a] + (x, y, z) — a ] 3 = 0 , 
<f> (€, y, £) 4- A = 0, 
4 (g, y, £) — a — f = 0 . 
The equation of the tangent plane at such a point is 
[2(p (g, r), £) + 4 (g, y, £) — «] 
+ {<£ {g, y, £) + 3 [i/z (g, rj, £) — a] 3 } 
This reduces to 
C(f> | /A7 . N 00 , /r7 
(X - + (Y - vPi + (Z - = 0. 
(X-0| + (Y-,)| + (Z-C)| 
0 </> 
= 0 . 
which is equivalent to 
c 
(X - f)s+ (Y - v)g + (z - £) 1 } [<Mf, £) + *] = 0 . 
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