242 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
which always passes through a fixed line ; there are thus on the reciprocal surface 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 discrimi- 
nant 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, xf known a 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 
I s — 27J 2 =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 curve is obtained as a curve 
=0, without 
or with further speciality. 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 , w) be A, B, C, D, then the surface is 
(AD - BC) 2 — 4(AC - B 2 )(BD - C 2 ) = 0, 
having the cuspidal curve 
of a thrown out factor. 
A, B, C 
B, C, D 
= 0, subject however to modification in the case 
Explanation as to the Sections of the Memoir . . Article Nos. 33 & 34. 
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 consequently 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 
* In some easy cases, for instance XYI=12— 4C 2 , tlie equation of the reciprocal surface is obtained other- 
wise by a direct elimination. 
t 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. 
