AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
239 
(C.) The Locus 
k£p (ay — (3 2 ) -j- kQ l ~ l ( ae~y~ — 2/38exy 4~ yS^x 2 ) 
— £ (ah 2 — 2/3gh + yg 2 ) — (hSx — gey) 2 = 0 
is an Ordinary Envelope. 
The condition uv — w 2 = 0, 
£ [(ay — /3 2 ) £ + (aeh/ 2 — 2/3 8eXy + y S’x 3 )] = 0, 
is not fulfilled at every point of this locus. 
To prove that it is an envelope it will be sufficient to show that if x, y, £, be chosen 
so that 
(a£ *T S^x 2 ) ( x — a) + (/3£ + Sexy) (y — h) + y£ = 0, 
(P£ + (x — a) + (y£ + ehy 2 ) (y — b) + h£ = 0, 
</£ (x — a) + lit, (y — b) + K n = 0 , 
then the surface 
ktp (ay — /3 2 ) -j- k£ ll ~ x (ae 2 y 2 — 2/38exy -j- yS 2 x 2 ) 
— £ (ah 2 — 2 /3gli + yg 2 ) — (hSx — yey) 2 = 0 
touches the surface 
(a£ + S 2 x 3 ) (x — a) 2 + 2 (f3i + Sexy) (x — a) (y — h) + (y£ + e 3 y 2 ) (y — 6) 3 
+ 2y£ (x — a) + 2A£ (y — fr) + &£" = 0. 
Calling the last two equations <f>= 0 ,f= 0 respectively, the conditions for contact 
may be expressed thus. 
The same values of x, y, £, must satisfy 
P — ffi /= 0 
/ ny / p/ = / d/ 
Ite / Da- Dy / Dy D? / I>£ ’ 
where x, y, £ are the independent variables. 
The values chosen for x, y, £ obviously make f = 0. 
Also eliminating x — a, y — 6, the result is £ 2 <£ = 0. 
Hence the values of X, y, £ can be chosen so as to make </> = 0. 
Next 
?/ 
D.r 
23 (x — a) [Sx (x — a) + ey (y — 6)] 
+ 2 [(a£ + S 2 x 2 ) (x — a) + (/3£ + Sexy) (y — 8) + y£] 
= 2 S(x a) [Sx (x — a) -f- ey (y — 6)] 
for the above values of x, y, £. 
