PROFESSOR CAYLEY ON CUBIC SURFACES. 
2 82 
W: rite 
A=±x+12cz,. 
B = 6cx— oby-\-Qbdz—2aw, 
C = Qbdx—Aady-\-Aad 2 z-\-Z(?>b 2 —Aac)w, 
and therefore 
L — y 2 + Aw, 
M=2dxy+2Bw, 
N = -4<ZV-4Cw, 
then we have 
P = ~ { — (y 2 + Aw)(4 d 2 x 2 + 4C w) + (2 dxy + 2Bw) 2 } 
— — 4 { Cy 2 — 2Bdxy + Ad 2 x 2 + w(AC— B 2 ) } , 
or the equation is 
4L 2 { Cy 2 - 2Bdxy+ Ad 2 x 2 + w(AC- B 2 ) } 
+ 182LMN+162M 3 + 27^wN 2 =0. 
101. Consider the'section by the plane w=0, we have ~L=y 2 , AA-=.2dxy, N= — 4 d 2 x 2 , 
and the equation becomes Ay\Cy 2 — 2Bdxy-A-Ad 2 x 2 )-\-(A2% — 144=)— 16^Vy 3 2:=0 ; 
which substituting for A, B, C the values 
A =4^+1 2 cz, 
B = 6 cx— 3 by-\-ebdz, 
C = bbdx— Aady-\- Aad 2 z, 
becomes A6dy 3 (y—dz)(dx 3 —3cx 2 y-\-3cxy 2 —ay 3 )=0 ; which is in fact the line w= 0, y— 0 
(reciprocal of the edge) three times, and the lines w= 0, (y—dz)(d, —c, b, — djx,y ) z =. 0 
(reciprocals of the biplanar rays) each once. Observe that the edge (X=Q, Z=0) is not 
a line of the cubic surface, but the reciprocal line y— 0, w= 0 presents itself as an oscular 
line of the reciprocal surface. 
102. The equations of the cuspidal curve are in the first instance obtained in the form 
Consider the two equations 
L, M, 3N 
12 zw, L , — 4M 
= 0 . 
L 2 -12zwM=0, 
LM+ 9zwN=0, 
each of the fourth order, but which are satisfied by zw= 0, L=0; that is, by 
(w=0, y 2 =0), (2=0, y 2 -\-Axw=0). The line (w= 0, y= 0) however presents itself in 
the intersection of the two surfaces, not twice only, but 4 times. To show this, observe 
that the line in question is a nodal line on the surface L 2 — 122 wM =0 ; in fact, attending 
only to the terms of the second order in y, w, we find 
{(Ax-\-A 2 cz) 2 — 1 44 cxz —144 bdz 2 }w 2 —2Adxzyw=0, 
