444 
A MEMOIR ON CUBIC SURFACES. 
[412 
The complete intersection with the cubic surface is made up of X = 0, Y = 0 
(BB-a,x\s) four times, of Y = 0, Z= 0 and F = 0, 1T=0 (CJS-axes) each three times; and 
of a residual conic, which is the spinode curve ; a' = 2. The equations of the spinode 
curve are F 2 — 3ZTT = 0, 4X+3F=0; viz. it lies in a plane passing through the 
BB-axis; since there is no facultative line, /3' = 0. 
Reciprocal Surface. 
184. The equation is found to be 
or say this is 
(if- + 4 zw) 2 — xy 3 — ‘S6xyzw + 27 sc 2 zw = 0, 
lQz 2 w 2 + (8y 2 — 3Qxy + 27x 3 ) zw + y 3 (y —x) = 0. 
The section by plane w=0 (reciprocal of B 3 = D) is w — 0, y 3 (y — x) = 0, viz. this is 
the line w — 0, y = 0 (reciprocal of edge) three times, and the line w = 0, y — x = 0 
(reciprocal of ray) once ; and the like as to section by plane z = 0. 
The section by plane x = 0 (reciprocal of C 2 = A) is x = 0, (y 2 + 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\y 2 — 12zw, 9# — 8y) 2 = 0, 
where 4.1.3zw — y 2 = — (y 2 — 12^y) ; and there is thus a cuspidal conic y 2 — 12zw = 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, to = 0 (reciprocal of the common biplane) is a 
binode of the surface; or there is the singularity B' = 1.