AND LINKS IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 205 
Example 8.— Envelope Locus, each Surface of the System having Stationary Contact 
with the Envelope. 
Let. the surfaces be 
a{x — a) 3 + 0 {y — bf + 3 [y (a? — a) + 8 (y — b)J + 2 2 — c 2 = 0, 
where a, /3, y, 8, c are fixed constants ; a, b the arbitrary parameters. 
(A.) The Discriminant, 
It is the same as that of the equation 
aX 3 + /3Y 3 + 3Z ( 7 X + SY) 2 + (2 2 - c 2 ) Z 3 = 0. 
Therefore 
S = a/3yS ( z~ — c~), 
T = (z 3 - c 3 ) [a 2 /3 2 (z 2 - C 3 ) + 4 (aS 3 - /3y 3 ) 3 ]. 
Therefore 
A = (z 3 — c 3 ) 3 [{a 2 /3 2 (z 3 — c 2 ) -j- 4 (aS 3 — /3y 3 ) 3 ) 3 + 64a 3 /3 3 y 3 S 3 (z 3 — c 2 )]. 
(B.) The Envelope , swcA that each Surface has Stationary Contact with it, is z~ — c~ = 0. 
Transform the equation by means of the equations £C = « + X, y=bfY, 2=io-f Z, 
and it becomes 
aX 3 + /3Y 3 + 3 (yX + SY) 2 + Z 3 ± 2cZ = 0. 
The tangent plane at the new origin is Z = 0 ; it cuts the surface in the curve 
aX 3 + /3Y 3 + 3 (yX + SY) 2 = 0 } 
which has a cusp at the origin. 
Hence the contact is stationary. 
Hence the factor (z 3 ~~ c 3 ) 2 in the discriminant is accounted for. 
(C.) The Locus 
[a 3 /3 3 (z 3 - c 3 ) + 4 (*S 3 - /3y 3 ) 2 } 2 + 64a 3 /3 3 y 3 S 3 (z 3 - C 3 ) = 0 
is an ordinary envelope. 
This may be proved by finding the tangent planes parallel to the plane 2=0. 
It is necessary to satisfy at the same time 
a (a _ a f + /3(y - 5) 3 + 3 [y (x - a) + S (y - 6)] 3 + ** - c 3 = 0 . (110) 
a (# — a) 2 -T 2y [y (it — a) -j- 3 (y — 6) | — 0 
S (y 3 ) 2 -h 28 |y (<£ " «) -j- 8(y — 0)] - o (HI)* 
22 zh 0 
