349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
315 
centre, which is not a mere node but a cusp on the cubic, or instead of a merely 
nodal cubic we have a cuspidal cubic; and corresponding to the one-with-twofold value 
of k we have a one-with-twofold critic centre, being of course a mere node on the nodal 
cubic. 
5. In the case in question of a twofold and one-with-twofold value of k, the line 
Xx + ¡iy + vz = 0, or say the satellite line, envelopes a curve which might be termed the 
twofold and one-with-twofold envelope, but which is spoken of simply as the envelope. 
The locus of the twofold centre is a curve which is called the twofold centre locus. 
The locus of the one-with-twofold centre is a curve which is called the one-with- 
twofold centre locus. 
These definitions premised, the following results may be stated; 
fi. The equation in 0 may be represented in the three equivalent forms 
1 _JL 1_ _2 
0+ \ Jr 0 + 6 + V 0 
X /Lt V 1 A 
0 + X ^ 0 + /X 0 + v 
0 s — 0 (yv + v\ + Xyu,) — 2\/jlv = 0. 
7. The critic value of k and the coordinates of the critic centre are then given 
by the equations 
k = 
{0 + X) (0 + yti) ( 0 + V) 
1 1 
x : y : z : x + y + z : \x + /iy + vz = 
0 + \ ' 0 + fX ’ 0 + v ' 0 
: I : 1. 
8. The condition for a twofold and one-with-twofold value of k is 
X 3-f-yti 3-f-Z^ i — 0, 
or, what is the same thing, 
(yv + vX + X/a) 3 — 2 7X 2 yaV = 0, 
which equations may either of them be considered as the line-equation of the envelope. 
The equation in the coordinates (x, y, z), or point-equation of the envelope is 
\/ x + v 7 y + v 7 z — 0, 
or, in its rationalised form, 
— 4 (yz 3 + y A z + zx 3 + z 3 x + xy 3 + x 3 y) 
+ 6 (y 2 z 2 + z-x 1 + x?y-) — 124 (x-yz + xy-z + xyz~) = 0. 
40—2 
+ y A + &