390 
A MEMOIR ON CUBIC SURFACES. 
[412 
56. In explanation of the discussion of the reciprocal surface, it is convenient to 
remark that we have 
Node C 2 , X = 0, Y= 0, Z = 0. 
Tangent cone is 
(a, b, c, f g, h\X, Y, Z) 2 = 0. 
Nodal rays are sections of cone by planes 
X = 0, Y= 0, Z = 0 respectively, viz. equa 
tions of the rays are 
X = 0, bY 2 + 2/YZ + cZ 2 = 0, 
F = 0, cZ 2 + 2gZX + aX 2 = 0, 
Z =0, aX 2 + 2hXY + bY 2 = 0. 
57. 
conic 
viz. the lines 
Reciprocal plane is w = 0. 
Conic of contact is 
(A, B, G, F, G, H'fyoc, y, zf= 0, w = 0. 
Lines are tangents of this conic from points 
{y = 0, z = 0), {z = 0, x = 0), O = 0, y = 0) 
respectively, viz. equations are 
w = 0, cy 2 — 2fyz + bz 2 = 0, 
w = 0, az 2 — 2 gzx + coc 2 =0, 
w = 0, bx 2 — 2 hxy + ay 2 = 0. 
The equation shows that the section by the plane w = 0 is made up of the 
{A, B, G, F, G, H\x, y, z) 2 = 0, twice, and of the six lines, tangents to this conic, 
w — 0, cy 2 — 2fyz + bz 2 — 0, 
w = 0, az 2 — 2gzx + ex 2 = 0, 
tv — 0, bx 2 — 2 hxy + ay 2 = 0, 
each once; the lines in question (reciprocals of the nodal rays) are thus mere scrolar 
lines on the reciprocal surface. 
58. I do not attempt to put in evidence the nodal curve of the surface; by 
what precedes it is made up of 15 lines, intersecting 3 together in 15 points; and 
if we denote the six tangents of the conic just referred to by 
1, 2, 3, 4, 5, 6, 
then the fifteen lines are respectively lines passing through the intersections of each 
pair of these tangents; viz. through the intersection of the tangents 1 and 2, we have 
a line 12; and so in other cases; that is, the 15 lines are 12, 13.... 56. The lines 
12 and 34 meet; and the lines 12, 34, 56 meet in a point; we have thus the 15 
points 12.34.56, triple points of the nodal curve. 
59. As regards the cuspidal curve, the equation of the surface may be written 
(L 2 -12 kwM) (4 M 2 + 3 LN) - (LM + 9 kivN) 2 
= 3 (DM- + DN - ISkivLMN - 1 GJcwM 3 - 21k 2 w 2 N 2 ) = 0, 
and we thus have 
4 M 2 + 3 LN = 0, 
LM + 9JcwN = 0, 
L 2 — 12k wM = 0, 
or, what is the same thing, 
L , 
12M, 
-9 N 
kw, 
L, 
M 
(equivalent to two equations) for the equations of the cuspidal curve. Attending to 
the second and third equations, the cuspidal curve may be considered as the residual 
intersection of the quartic and quintic surfaces L 2 — 12kwM = 0, LM + 9kwN = 0, which 
partially intersect in the conic w = 0, L = 0; or say it is a curve 4x5 — 2; c' =18.