360
A MEMOIR ON CUBIC SURFACES.
[412
IV
= 12 — 2 C 2 ,
WXZ+ Y*(ryZ + 8W) + (a > b, c, d^X, Y) 3 = 0,
V
= 12 — B 4 ,
WXZ + (X + Z)(Y 2 - aX 2 - bZ 2 ) = 0,
VI
= 12-B 3 -C 2 ,
WXZ + Y 2 Z + (a, b, c, d^X, V) 3 = 0,
VII
= 12-B 5 ,
WXZ + Y J Z + YX 2 - Z 3 = 0,
VIII
= 12 — 30,,
Y 3 + Y 2 (X + Z + TV) + 4 aXZW= 0,
IX
= 12-2 B 3 ,
WXZ+(a, b, c, d^X, Y) 3 = 0,
X
= 12-B 4 -C 2 ,
WXZ + (X + Z) (F 2 - X 2 ) = 0,
XI
= 12 —B 9 ,
WXZ + Y 2 Z + X 3 -Z 3 = 0,
XII
= 12 - U„
W (X + Y + Z) 2 + XYZ = 0,
XIII
= 12-B 3 -2G 2 ,
WXZ+ Y 2 (X + Y + Z) = 0,
XIV
= 12 — fig — C 2 ,
WXZ+ Y 2 Z+ YX 2 = 0,
XV
= 12- U 7 ,
WX 2 + XZ 2 + Y 2 Z= 0,
XVI
= 12-4 C 2 ,
W(XY+ XZ + YZ) + XYZ= 0,
XVII
= 12-2 B 3 -C 2 ,
WXZ + XY 2 + Y 3 = 0,
XVIII
= 12 — B 4 — 2C 2 ,
WXZ + (X + Z) Y 2 = 0,
XIX
= 12-B 6 -C 2 ,
WXZ+ Y 2 Z + X 3 = 0,
XX
= 12- U 9i
WX 2 + XZ 2 + Y 3 =0,
XXI
= 12-3 B„
WXZ + Y 3 = 0.
XXII
= 3, >8(1, 1),
WX 2 + ZY 2 = 0,
XXIII
= 3, >8(1, 1),
X(WX+ YZ)+Y 3 = 0;
2. Where C 2 denotes a conic-node diminishing the class by 2; B 3 , B 4 , B 5 , B 6 a
biplanar node diminishing (as the case may be) the class by 3, 4, 5, or 6 ; and
U 6 , TJ 7 , U 8 a uniplanar node diminishing (as the case may be) the class by 6, 7, or 8.
The affixed explanation, which I shall usually retain in connexion with the Roman
number, shows therefore in each case what the class is, and also the singularities which
cause the reduction: thus XIII = 12 — B 3 — 2G 2 indicates that there is a biplanar node,
B 3 , diminishing the class by 3, and two conic-nodes, C 2 , each diminishing the class
by 2 ; and thus that the class is 12 — 3 — 2.2, = 5. As regards the cases XXII and
XXIII, these are surfaces having a nodal right line, and are consequently scrolls, each
of the class 3, viz. XXII is the scroll S( 1, 1) having a simple directrix right line
distinct from the nodal line, and XXIII is the scroll >8(1, 1) having a simple directrix
right line coincident with the nodal line: see as to this my “ Second Memoir on
Skew Surfaces, otherwise Scrolls,” Phil. Trans, vol. CLiv. (1864), pp. 559—577, [340].
3. The nature of the points C 2 , B 3 , B 4 , B 5 , B 6 , U 6 , U T , U 8 requires to be explained.
C (= C 2 ) is a conic-node, where, instead of the tangent plane, we have a proper
quadric cone.
B (= B 3 , B 4 , B 5 or B 6 ) is a biplanar-node, where the quadric cone becomes a plane-
pair (two distinct planes): the two planes are called the biplanes, and their line of
intersection is the edge :
In B 3 , the edge is not a line on the surface—in the other cases it is; this
implies that the surface is touched along the edge by a plane, viz. in B 4 , B 3 the
edge is torsal, in B 6 it is oscular:
