370 
A MEMOIR ON CUBIC SURFACES. 
[412 
series of such conics. It would be very instructive and interesting to carry out the 
investigation in detail. 
32. The equation of the reciprocal surface is found by equating to zero the 
discriminant of a ternary or a binary function^), viz. this is a ternary cubic, or a 
binary quartic, cubic, or quadric. The equation as given in the form disct. = 0, contains 
a factor which for the adopted forms of equations is always a power or product of 
powers of w, z, x( 1 2 ) known d priori, and which is thrown out without difficulty, the 
equation being thereby reduced to the proper order. There is the singular advantage 
that the process puts in evidence the cuspidal curve of the resulting reciprocal 
surface, viz. for a ternary cubic, the form obtained is S 3 —T' 2 = 0, and for a binary 
quartic it is the equivalent form 7 3 — 27 t / 3 = 0; but for the factor thrown out as just 
mentioned, we should have simply (S = 0, T = 0), or, as the case may be, (1 = 0, J= 0) 
for equations of the cuspidal curve; the existence of the factor occasions however a 
modification, viz. the intersection of the two surfaces is not an indecomposable curve, 
and the cuspidal curve is in most cases, not the complete intersection, but a partial 
intersection of the two surfaces. In several cases it thus happens that the cuspidal 
P, Q, P 
P\ QT, R' 
Similarly when the equation of the reciprocal surface is obtained by means of a 
binary cubic; if the coefficients hereof (functions of course of the coordinates x, y, z, iv) 
be A, B, G, D, then the surface is 
curve is obtained as a curve 
= 0, without or with further speciality. 
having the cuspidal curve 
of a thrown out factor. 
(AD - BG) 2 — d(AG — B 2 ) (BD - G 2 ) = 0, 
A, B, G 
B, G, D 
= 0, subject however to modification in the case 
Article Nos. 33 and 34. Explanation as to the Sections of the Memoir. 
33. As regards the following Sections I to XXIII, it is to be observed that for 
the general surface 1 = 12, I do not attempt to form the equation of the reciprocal 
surface, and in some of the other cases, 11 = 12 — C 2 &c., the equation of the reciprocal 
surface is either not obtained in a completely developed form, or it is too complicated 
to allow of its being dealt with, for instance so as to put in evidence the nodal 
curve of the surface. Portions of the theory given in the latter sections are con 
sequently omitted in the earlier ones, and in particular in the Section I there is 
given only the diagram of the 27 lines and the 45 planes (with however developments 
as to notation and otherwise which have no place in the subsequent sections), and 
with the analytical expressions for the several lines and planes, although from the 
1 In some easy cases, for instance XVI = 12 -4C 2 , the equation of the reciprocal surface is obtained other 
wise by a direct elimination. 
2 The factor is in general a power or product of powers of the linear functions which, equated to zero, 
give the equations of the planes reciprocal to the several nodes of the surface.