316 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
Reciprocal Surface. 
184. The equation is found to be 
[if + 4 zwf — xif — 3 6 xyzw + 27 x*zw =0, 
or say this is 
16zW4-(8?/ 2 — Sftxy+'ORx^zw+y^y— a?)=0. 
The section by plane w = 0 (reciprocal of B 3 =D) is w= 0, y\y — x)= 0, viz. this is the 
line w= 0, y = 0 (reciprocal of edge) 3 times, and the line w=(), y—x — 0 (reciprocal of 
ray) once ; and the like as to section by plane z= 0. 
The section by plane #=0 (reciprocal of C 2 =A) is x=0, 4;zw) 2 =0, viz. this is 
the conic (reciprocal of nodal cone) twice. 
There is no nodal curve ; b'= 0. 
185. Cuspidal curve. The equation of the surface may be written 
(1, —y, 3zw T fy 2 —12zw, 9x—Syf=0, 
where 4 . 1 . 2>zw— y 2 = — (y 2 — 12zw) ; and there is thus a cuspidal conic y 1 — 12^w=0, 
9x— 8y=0 : wherefore c'=2. 
Attending only to the terms of the second order in y, z, w, the equation becomes 
x 2 zw=0 ; that is, the point y=0, z= 0, w = 0 (reciprocal of the common biplane) is a 
binode of the surface ; or there is the singularity B'=l. 
