316 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
9. The equation of the twofold centre locus is 
\lx + *Jy + \lz = 0, 
nr, in its rationalised form, 
x 2 + y 2 + z 2 — 2 yz — 2 zx — 2 xy = 0; 
the curve is therefore a conic, and it may be spoken of as the twofold centre conic. 
10. The equation of the one-with-twofold centre locus is 
a? + y 3 + z 3 — (yz 2 + y 2 z + zx 2 + z 2 x 4- xy 2 + x 2 y) + 3 xyz = 0, 
the curve is therefore a cubic, and it may be spoken of as the one-with-twofold 
centre cubic. 
11. The before-mentioned equation AT * + yT * + v~^ — 0 is satisfied by 
X : fi : v = a -3 : /3 -3 : 7 -3 , 
where a 4- /3 + 7 = 0, and it is very convenient to introduce these quantities a, /3, 7 into 
the formulae. 
12. The equation of the satellite line giving a twofold and one-with-twofold centre is 
xyz. 
- 3 + TT 3 H—3 — 0; 
a 3 /3 3 7 3 
the coordinates of the point of contact with the envelope are x : y : z = a* : ¡3* : y 4 . 
The equation in 6 gives 6 1 = 0 2 = — for the values corresponding to the twofold 
2 
centre; and 9 3 = for the value corresponding to the one-with-twofold centre. 
The coordinates of the twofold centre, or cusp, are x : y : z = a 2 : ¡3 2 : y 2 . 
The coordinates of the one-with-twofold centre, or node, are 
x : y \ z = a 2 (/3-y) : ¡3 2 (y-a) : y^z-fi). 
The equation of the tangent at the cusp is 
(£ —7)^ + (7- a )| + 0—/3)- = 0. 
a P 7 
The equation of the line joining the cusp and the node, which line is also one of 
the tangents at the node is 
x y z 
- + 3 + - = 0. 
a /3 7 
The equation of the other tangent at the node is 
(2/3 7 + of) 2 + (2 7 a + /3») I + (2a/3 + ,») - = 0. 
a 7