The collected mathematical papers of Arthur Cayley.

198

ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.

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point in 5-dimensional space. It may be remarked that in this 5-dimensional space we
have the onefold cubic locus abc - f2 - bg2 -ch2 + 2fgh = 0, which is such that to any
position of the parametric point upon it there corresponds not a proper conic but a
line-pair; this may be called the discriminant-locus. We have also the threefold locus
the relation of which is expressed by the six equations
(bc-f=O0, cca-g2=, ab-h2=0, gh-af=0, hf-bg=0, fg-ch =0),
which is such that to any position of the parametric point thereon, there corresponds
not a proper conic but a coincident line-pair. I call this the Bipoint-locus(l), and
I notice that its order is =4; in fact to find the order we must with the equations
of the Bipoint combine two arbitrary linear relations,
(* a, b, c, f, g, h) = 0,
( a', b, c, f, h)g, =0;
the equations of the locus are satisfied by
a: b: c: h=: g: g: -= a2 y: ya: a3
(where a: /: y are arbitrary); and substituting these values in the linear relations,
we have two quadric equations in (a, /3, y), giving four values of the set of ratios
(a: /3: y); that is, the order is = 4, or the Bipoint is a threefold quadric locus.
17. The discriminant-locus does not in general present itself except in questions
where it is a condition that the conic shall have a node (reduce itself to a line-pair);
thus for the conies which have a node and touch a given curve (m, in), or, what is the
same thing, for the line-pairs which touch a given curve (m, in), the parametric point is
here situate on a twofold locus, the intersection of the discriminant-locus with the contact-locus. It may be noticed that this twofold locus is of the order 3 (n+ 2m), but
that it breaks up into a twofold locus of the order 3n, which gives the proper solutions;
viz. the nodal conies which touch the given curve properly, that is, one of the two
lines of the conic touches the curve; and into a twice repeated twofold locus of the
order 3m which gives the special solutions, viz. in these the nodal conic has with the
given curve a special contact, consisting in that the node or intersection of the two
lines lies on the given curve. By way of illustration see Annex No. 2. But the consideration of the Bipoint-locus is more frequently necessary.
18. Suppose that the conic satisfies the condition of touching a given curve; the
parametric point is then situate on a onefold contact-locus (a, b, c, f g, h)v =0 (to fix
the ideas, if the given curve is of the order m and class n, then the order q of the
contact-locus is =n + 2n). The contact-locus of any given curve whatever passes
through the Bipoint-locus; in fact to each point of the Bipoint-locus there corresponds
a coincident line-pair, that is, a conic which (of course in a special sense) touches the
given curve whatever it be; and not only so, but inasmuch as we have a special

1 In framing the epithet Bipoint, the coincident line-pair is regarded as being really a point-pair: see
post, No. 30.