368 A MEMOIR ON CUBIC SURFACES. [412 
Article Nos. 27 to 32. On the Determination of the Reciprocal Equation. 
27. Consider in general the cubic surface (*]£X, Y, Z, W) 3 = 0, and in connexion 
therewith the equation Xx + Yy + Zz + Ww = 0, which regarding therein X, Y, Z, W 
as current coordinates, and x, y, z, w as constants, is the equation of a plane. If 
from the two equations we eliminate one of the coordinates, for instance W, we obtain 
Yw, Zw, — (Xx + Yy + Zz)) 3 = 0, 
which, (X, Y, Z) being current coordinates, is obviously the equation of the cone, vertex 
(X = 0, F= 0, Z— 0), which stands on the section of the cubic surface by the plane. 
Equating to zero the discriminant of this function in regard to (X, Y, Z), we express 
that the cone has a nodal line; that is, that the section has a node, or, what is the 
same thing, that the plane xX + yY + zZ + wW = 0 is a tangent plane of the cubic 
surface; and we thus by the process in fact obtain the equation of the cubic surface 
in the reciprocal or plane coordinates (x, y, z, w). Consider in the same equation 
x, y, z, w as current coordinates, (X, Y, Z) as given parameters, the equation represents 
a system of three planes, viz. these are the planes xX + yY + zZ -f- wW' = 0, where W 
has the three values given by the equation (*$X, F, Z, W') 3 = 0, or, what is the same 
thing, X, Y, Z, W' are the coordinates of any one of the three points of intersection 
of the cubic surface by the line ~ ~ ^; (X, Y, Z, W’) belongs to a point on the 
surface, and 
xX + yY + zZ +wW' = 0 
is the polar plane of this point in regard to a quadric surface X- + Y- + Z- + W 2 = 0; 
the equation 
(*^Xw, Yw, Zw, — (Xx -\-Yy + Zz)) 3 = 0 
is thus the equation of a system of 3 planes, the polar planes of three points of the 
cubic surface (which three points lie on an arbitrary line through the point x = 0, 
y — 0, z — 0). In equating to zero the discriminant in regard to (X, Y, Z), we find the 
envelope of the system of three planes, or say of a plane, the polar plane of an 
arbitrary point on the cubic surface,—or we have the equation of the reciprocal 
surface, being, as is known, the same thing as the equation of the cubic surface in 
the reciprocal or plane coordinates (x, y, z, w). In what precedes we have the 
explanation of an ordinary process of finding the equation of the reciprocal surface, 
this equation being thereby given by equating to zero the discriminant of a function 
(*][X, F, Z) 3 , that is, of a ternary cubic function. 
28. The process, as last explained, is a special one, viz. the position of a point 
on the surface is determined by means of certain two parameters, the ratios X : Y : Z 
which fix the position of the line joining this point with the point (x = 0, y = 0, 
z = 0). More generally we may consider the position of the point as determined by 
means of any two parameters; the equation of the polar plane then contains the two 
parameters, and by taking the envelope in regard to the two parameters considered as 
variable, we have the equation of the reciprocal surface.