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A MEMOIR ON CUBIC SURFACES. 
369 
29. But let the parameters, say 0, <£, be regarded as varying successively ; if 
alone vary, we have on the surface a curve ©, the equation whereof contains the 
parameter 9, and when 6 varies this curve sweeps over the surface. The envelope in 
regard to cfi of the polar plane of a point of the surface is a torse, the reciprocal 
of the curve ©, and the envelope of the torse is the reciprocal surface. In particular 
the curve © may be the plane section by any plane through a fixed line, say, by 
the plane P — 9Q = 0 ; the section is a cubic curve, the reciprocal is a sextic cone 
having its vertex in a fixed line (the reciprocal of the line P = 0, Q = 0), and the 
reciprocal surface is thus obtained as the envelope 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 6 ; or writing 9 = P : Q, we obtain the equation of the reciprocal 
surface by equating to zero the discriminant of a binary 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 © is then a conic, its reciprocal is a quadricone, and the envelope 
of this quadricone is the required reciprocal surface. This is really what Schiaffi 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 fines 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 fine made use of. But 
this is an investigation which I do not enter upon : I adopt in each case Schlaffi’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 9 from the equations of the curve. In particular let the torse be the 
circumscribed cone having its vertex at any point of a fixed fine ; the reciprocal 
figure is then a plane curve, the plane of which passes through the fine which is 
the reciprocal of the fixed fine ; it is moreover clear that if the position of the vertex 
on the fixed fine be determined by the parameter 9 linearly (for instance if the 
vertex be given as the intersection of the fixed fine by a plane P — 0Q = 0), then 
the equation of the plane of the curve will be of the form P' = 9Q', containing the 
parameter 0 linearly ; the other equation of the plane curve will contain 0 rationally, 
and the elimination will be at once effected by substituting in this other equation 
for 9 its value, = P' 4- Q'. And observe moreover that if the fixed fine be a fine 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 
which always passes through a fixed fine ; there are thus on the reciprocal surface 
c. vi. 47