i the 
that 
the 
parameter p in the equation V + pz-s = 0, which are such as to give rise to a nodal 
curve. It is to be noticed that except for the Hyperbolas and for the Cubical 
Parabola, the satellite s = 0 is either a line passing through a given point at infinity 
(determinate, that is, as regards its direction), or in the case of the Trident Curve 
it is the given line infinity; there is at most only a single series of positions to be 
considered, and the theory is a short and easy one ; and for the Cubical Parabola, 
although the satellite line 5 = 0 is here an arbitrary line, yet on account of the cusp 
at infinity, there is not any critic value of p, or in fact any distinction of cases. 
The only other case where the satellite line s = 0 is an arbitrary line, admitting 
therefore of a double series of positions, is that of the Hyperbolas; and the division 
into groups constitutes an extensive and interesting theory, which is insufficiently 
discussed by Pliicker ; and it was with a view to the development of this theory 
that my Memoir, On a Case of the Involution of two Cubic Curves, (ante, pp. 39 to 81), 
referred to in the sequel as Memoir on Involution, [349], was written. I remark that 
of the three curves there established as material to the theory, and which are further 
spoken of in the sequel of the present memoir, viz. the envelope, the twofold centre 
locus, and the one-with-twofold centre locus, Pliicker considers only the twofold centre 
locus. I proceed to apply the results of that Memoir to the present theory. 
As to the Groups of the Hyperbolas A. Article Nos. 51 to 53. 
51. The assumed form of equation was pqr + pz^s = 0, but using now 
x = 0, y = 0, z — 0 
(instead of p — 0, q = 0, r=0) for the equations of the asymptotes, we may imagine 
the implicit constants so determined that the line infinity (before represented by 
z — 0) shall have for its equation x + y -\- z = 0; writing moreover \x + py + vz — 0 for 
the satellite line s = 0, and Jc in the place of p, the equation becomes 
xyz -{-Jc (x + y + zf (Xx + py + vz) = 0, 
which is the form considered in the Memoir just referred to. 
52. It is there shown that for an arbitrary position of the satellite line, the 
parameter Jc, or, what is the same thing, the auxiliary parameter 6, may be determined 
by a cubic equation in such manner that the curve shall have a node; the node, or 
rather the site of the node is termed a critic centre; and there are consequently 
three critic centres (all real or else one real, two imaginary). If however the satellite 
line touches a certain curve called the envelope, then two of the critic centres unite 
together, forming a twofold centre which is (not a mere node but) a cusp on the 
corresponding cubic curve; the other critic centre is termed a one-with-twofold centre; 
c. v. 47