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PROFESSOR CAYLEY ON CUBIC SURFACES. 
VIII =12-3C a , Y 3 +Y 2 (X+Z+W)+4aXZW=0, 
IX =12 — 2B 3 , WXZ+(a, b, c, dJX, Y) 3 =0, 
X =12 — B 4 — C 2 , WXZ+(X-fZ)(Y 2 -X 2 )=0, 
XI =12 — B 6 , WXZ+Y 2 Z+X 3 -Z 3 =0, 
XII =12 -U 6 , W(X+Y+Z) 2 +XYZ = 0, 
XIII =12— B 3 — 2C 2 , WXZ+Y 2 (X+Y+Z)=0, 
XIV =12— B 5 — C 2 , WXZ+Y 2 Z+YX 2 =0, 
XV =12— U 7 , WX 2 +XZ 2 +Y 2 Z=0, 
XVI =12 — 4C 2 , W(XY+XZ+YZ)+XYZ=0, 
XVII =12 — 2B 3 — C 2 , WXZ+XY 2 +Y 3 =0, 
XVIII =12-B 4 -2C a , WXZ+(X+Z)Y 2 =0, 
XIX =12 — B 6 — C 2 , 
XX =12— U 8 , 
XXI =12— 3B 3 , 
XXII = 3, S(l, 1), 
XXIII = 3, S(I7T), 
WXZ+Y 2 Z+X 3 =0, 
WX 2 +XZ 2 +Y 3 =0, 
WXZ+Y 3 =0, 
WX 2 +ZY 2 =0, 
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 , U 7 , U 8 a uni- 
planar node diminishing (as the case may be) the class by 6, 7, or 8. The affixed expla- 
nation, which I shall usually retain in connexion with the Roman number, shows there- 
fore in each case what the class is, and also the singularities which cause the reduction : 
thus XIIX = 12 — B 3 — 2C 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(l, 1) having a simple directrix right line distinct from the nodal line, and 
XXIII is the scroll S(l, 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. 
3. The nature of the points C 2 , B 3 , B 4 , B 5 , B 6 , U 6 , U 7 , 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 intersec- 
tion 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 5 the edge is torsal, in B s 
it is oscular : 
