396 
A MEMOIR ON CUBIC SURFACES. 
[412 
69. In the discussion of the equation it is convenient to write down the relations 
of the two surfaces, thus: 
Cubic surface. 
B 3 , X = 0, 7=0, Z=0 
Biplanes X + Y + Z = 0 
IX + mY + nZ — 0, 
intersecting in edge. 
Bays in first biplane, 
X = 0, Y+Z=0 ; 7=0, Z+X = 0, 
Z=0, X+Y = 0; 
rays in second biplane, 
X = 0, mY+nZ= 0; 7=0, nZ+lX= 0, 
Z = 0, IX + mY = 0. 
7 0. The equation puts in evidence the section by the plane w = 0, viz. this is 
the line a = 0 (reciprocal of the edge) three times, and the six lines (reciprocals of 
the rays) each once. Observe that the edge is not a line on the cubic; but its 
reciprocal is a line, and that an oscular line on the reciprocal surface; the six lines 
(reciprocals of the rays) are mere scrolar lines on the reciprocal surface; they pass, 
three of them, through the point x = y = z, and the other three through the point 
x : y : z = l : m : n; that is, they are six tangents of the point-pair (reciprocal of the 
pair of biplanes) formed by these two points. 
71. I do not attempt to put in evidence the nodal curve on the surface; by 
what precedes it consists of 9 lines, reciprocals of the mere lines. If we denote by 
1, 2, 3 and 4, 5, 6 the lines which pass through the points x = 0, y=0, z = 0 and 
through the point x : y : z— l : m : n respectively, then these intersect in the nine 
points 14, 15, 16, 24, 25, 26, 34, 35, 36; and through each of these there passes a 
nodal line which may be represented by the same symbol; that is, we have the nodal 
lines 14, ....36. Two lines such as 14, 25 meet; and three lines such as 14, 25, 36 
meet in a point; we have thus the six points 14.25.36 &c. triple points on the 
nodal curve ; as before, b' = 9, t' = 6. 
72, The cuspidal curve is given by the equations 
k 2 w 2 — 2kwt + a 2 , 24 (kwv + cryjr), — 36 (4Imnk xyziv — yjr 2 ) j = 0. 
kw , khu- — 2kivt + cr, 2 (kwv + a\fr) 
Writing down the two equations, 
(k 2 w 2 — 2kwt + a 2 ) 2 — 24kw (kwv + ayjr) = 0, 
(k 2 w 2 — 2kwt + <r 2 ) (kwv + o-\}r) + 18w (Imnk xyzw — \{r 2 ) = 0, 
these are respectively of the orders 4 and 5; but they intersect in the line w = 0, 
cr = 0 taken four times, or say, the cuspidal curve is a partial intersection 4.5 — 4; 
c' = 16. 
Reciprocal surface. 
Plane w = 0, 
Points in w = 0, viz. 
x = y —z and x : y : z = l : m : n, 
in line (m — n)x + (n — l) y + (l — m) z= 0, 
that is, Xx + /xy + vz — 0, or a = 0. 
Lines in plane w = 0, and through first 
point, viz. 
y — z = 0, z — x = 0, x — y = 0; 
lines through second point, viz. 
ny — mz = 0, nz — lx = 0, lx — my = 0.