320 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
or finally if 
AT 3 -f- fi~' A + v~ 3 = 0, 
so that the condition is satisfied if A = a 3 , y = (3 3 , v = <y 3 where a + (3 + 7 = 0. In 
fact with these values the equation in 6 becomes 
that is 
(aftyd) 3 — 3a/3y6 — 2 = 0, 
(a/3yd + iy(oil3yd-2) = 0, 
so that the twofold value is 6 X = 6, = : and the one-with-twofold value is 
aPY 
n = _2_ 
3 a/V 
23. It is throughout assumed that the quantities a, /3, 7 satisfy the condition 
a + /3 + y = 0. The result just obtained shows that the line 
x v z 
—I- — 4- - 
a 3 /3 3 7 3 
0, 
is a twofold and one-with-twofold satellite line. From this equation, considering a, ¡3. 7 
as variable parameters satisfying the condition a. + /3 + 7 = 0, we find at once the 
equation of the curve enveloped by the line in question, which curve is called simply the 
envelope—viz. the coordinates of the point of contact are found to be x : y : 2 = a 4 : /3 4 : y 4 , 
and thence the equation of the envelope is 
or rationalising, it is 
t/x + ZJy + tfz = 0, 
x x + y* + z i — 4 (yz 3 + y 3 z + zx A + z 3 x + xy 3 + x 3 y) 
+ 6 (jfz 2 + z 2 x 2 + x 2 y 2 ) — 124 (x 2 yz + y 2 zx -f z 2 xy) = 0. 
The before-mentioned equation A 3 + y 3 + v 3 = 0, or 
(yu.1/ + ^A + A//) 3 — 27 X 2 y 2 v 2 = 0, 
may be considered as the tangential equation; the envelope is thus of the order 4, 
and the class 6. 
24. It is easy to show that the curve has three nodes the coordinates whereof 
are (—4, 1, 1), (1, —4, 1), (1, 1, —4); and this being known, the equation may be 
transformed so as to put the nodes in evidence. I effect the transformation syntheti 
cally as follows, viz. writing x + y + ^ = a, yz + zx + xy = q, xyz — r, the equation of the 
curve is 
(a- 4 — 2 qa 2 + 2<? 2 + 4 ra) 
— 4 ( qa 2 — 2 q 2 — ru) 
+ 6 ( q 2 — 2ra) 
— 124 ( ra) = 0,