PROFESSOR CAYLEY ON CUBIC 'SURFACES. 
263 
(equivalent to two equations) for the equations of the cuspidal curve. Attending to 
the second and third equations, the cuspidal curve may he considered as the residual 
intersection of the quartic and quintic surfaces L 2 — 1 27avM = 0, LM -f- 9/fooN = 0, which 
partially intersect in the conic w= 0, L=0 ; or say it is a curve 4x5 — 2 ; d = 18. 
Section 111=12 -B 3 . 
Equation 2 W(X + Y+ Z)(ZX + mY +%Z) + 2£XYZ = 0 . Article Nos. 60 to 72. 
60. The system of lines and planes is at once deduced from that belonging to 
11=12 — C 2 , by supposing the tangent cone to reduce itself to the pair of biplanes; 
3 of the planes (a) of 11 = 12 — C 2 thus coming to coincide with the one biplane, and 
three of them with the other biplane. 
Lines. 
CO CO CO 
C5 Cfl 
fcO «> bo 
G5 C7i ^ 
S ^ 
C5 Cn rfx CO tt> — 
III= 
12— B 3 
51 
O 
X 
if 
to 
C5 
X 
co 
|l 
SSI 
CO 
123 
456 
2x6=12 
Biplanes. 
14 
• 
15 
* 
• 
16 
. 
24 
a 
Ph 25 
26 
9x3=27 
■ 
Biradial planes each con- 
taining a ray 1, 2, or 3 
of the one biplane, and 
a ray 4, 5, or 6 of the 
other biplane. 
34 
* 
35 
• 
1 
36 
14.25.36 
14.26.35 
15.26.34 
15.24. 36 
6x1 = 6 
• 
• 
• 
Planes each containing 
three mere lines. 
16.24.35 
16.25.34 
17 45 
. 
. 
Mere lines, in each 
biradial plane, 
one. 
Bays, 1, 2, 3 and 
4, 5, 6, in the 
two biplanes 
respectively. 
