PROFESSOR CAYLEY ON CUBIC SURFACES. 
241 
vary, we have on the surface a curve 0, the equation whereof contains the parameter 0, 
and when Q varies this curve sweeps over the surface. The envelope in regard to p of 
the polar plane of a point of the surface is a torse, the reciprocal of the curve 0, and the 
envelope of the torse is the reciprocal surface. In particular the curve 0 may be the 
plane section by any plane through a fixed line, say, by the plane P — 6 Q=0 ; the section 
is a cubic curve, the reciprocal is a sextic cone having its vertex in a fixed line (the reci- 
procal of the line P = 0, Q=0), and the reciprocal surface is thus obtained as the enve- 
lope of this cone ; assuming that the equation of the sextic cone has been obtained, this 
is an equation of a certain order in the parameter 0; or writing 0=P : Q, we obtain the 
equation of the reciprocal surface by equating to zero the discriminant of a Unary 
function of (P, Q). 
30. With a variation, this process is a convenient one for obtaining the reciprocal of a 
cubic surface : we take the fixed line to be one of the lines on the cubic surface ; the 
curve 0 is then a conic, its reciprocal is a quadricone, and the envelope of this quadri- 
cone is the required reciprocal surface. This is really what Schlafli does (but the 
process is not explained) in the several instances in which he obtains the equation of 
the reciprocal surface by means of a binary function. I remark that it would be very 
instructive, for each case of surface, to take the variable plane successively through the 
several kinds of lines on the particular surface ; the equation of the reciprocal surface 
would thus be obtained under different forms, putting in evidence the relation to the 
reciprocal surface of the fixed line made use of. But this is an investigation which I do 
not enter upon : I adopt in each case Schlafli’s process, without explanation, and merely 
write down the ternary or (as the case may be) binary function by means of which the 
equation of the reciprocal surface is obtained. 
31. It is to be mentioned that there is a reciprocal process of obtaining the equation 
of the reciprocal surface ; we may imagine, touching the cubic surface along any curve, a 
series of planes ; that is, a torse circumscribed about the surface, and the equation whereof 
contains a variable parameter 6 ; the reciprocal figure is a curve, the equations whereof 
contain the parameter 0 ; the locus of this curve is the reciprocal surface ; that is, the 
equation of the reciprocal surface is obtained by eliminating 6 from the equations of the 
curve. In particular let the torse be the circumscribed cone having its vertex at any 
point of a fixed line ; the reciprocal figure is then a plane curve, the plane of which 
passes through the line which is the reciprocal of the fixed line ; it is moreover clear 
that if the position of the vertex on the fixed line be determined by the parameter 6 
linearly (for instance if the vertex be given as the intersection of the fixed line by a 
plane P — 0Q=O), then the equation of the plane of the curve will be of the form 
P'=$Q', containing the parameter 6 linearly ; the other equation of the plane curve will 
contain Q rationally, and the elimination will be at once effected by substituting in this 
other equation for 6 its value, =P'-t-Q\ And observe moreover that if the fixed line 
be a line on the cubic surface, then the cone is a quadricone having for its reciprocal a 
conic ; the reciprocal surface is thus given as the locus of a variable conic, the plane of 
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