262 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
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,c,,hJX, 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, l Y 2 -f 2/ YZ + c Z 2 = 0, 
Y = 0, cZ 2 +2#ZX+aX 2 =0, 
Z=0, «X 2 +2AXY+5Y 2 =0. 
57. The equation shows that the section 
(A, B, C, F, G, IT^a\ y, z) 2 = 0, twice, and 
the lines 
w — 0, cy 1 — 
w= 0, az 2 —2t 
w= 0, lx 2 — 2} 
Reciprocal plane is W— 0. 
Conic of contact .is 
(A, B, C, F, G, HX^, y, z) 2 = 0,. w=0. 
Lines are tangents of this conic from points 
{y= 0, z=0), (z— 0, x=0), (x=0, y- 0) 
respectively, viz.. equations are 
w=0, cy 2 -2fyz + bz 2 =0, 
w=0 , az 2 — 2gzx + ex 2 = 0, 
w= 0, bx 2 -27ixy-\-ay 2 =0. 
»y the plane w = 0.is made up of the conic 
of the six lines, tangent to this conic, viz. 
yz + bz 2 — 0, 
zx-]-cx 2 =0, 
xy+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 be 
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 - 12Md)(4M 2 + 3LN) - (LM + 9£wN) 2 
= 3(L 2 M 2 + L 3 N — 1 8/rzoLMN - 16MT- 27/^WN 2 )=0, 
and we thus have 
4M 2 + 3LN =0, 
LM + 9MT=0, 
L 2 -12M1=0, 
L , 12M, — 9N 
Jew, L, M 
or, what is the same thing, 
1=0 
