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PROFESSOR CAYLEY ON CUBIC SURFACES. 
On the Determination of the Reciprocal Equation. Article Nos. 27 to 32. 
27. Consider in general the cubic surface (*^X, Y, Z, W) 3 =0, and in connexion 
therewith the equation Xx-\-Yy-\-Zz-\-Yifw=^, 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 
(*£Xw, Yw, Zw, -(Xx+Yy+Zz)) 3 =0, 
which, (X, Y, Z) being current coordinates, fis obviously the equation of the cone, vertex 
(X=:0, Y=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 ^X-f^Y-(-zZ-fwW=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 reci- 
procal 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-\- wW' = 0 , where W' has the three 
values given by the equation (#)£X, Y, 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 ^=y=^; (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 2 -{-Y 2 +Z 2 -J-W 2 =0 ■ 
the equation 
(*fXw, Yw, Zw, -(Xx+Yy + Zz)f=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, tv). 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 (ffX, Y, 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 (^=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 contams 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. 
29. But let the parameters, say 6, <p, be regarded as varying successively; if <p alone 
