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: