198 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
point in 5-dimensional space. It may be remarked that in this 5-dimensional space we 
have the onefold cubic locus abc — a/ 1 2 — bg 3 — ch 2 -4- 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' 2 = 0, ca - g 2 = 0, ab — h 2 = 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 (*), 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, 
(* $a, b, c, /, g, h) = 0, 
(*'$«, b, c, /, g, h) = 0; 
the equations of the locus are satisfied by 
a : b : c : / : g : h = a 2 : /3 2 : y 2 : /3y : ya : a/3 
(where a : ¡3 : <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 : 7); 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 conics which have a node and touch a given curve (m, n), or, what is the 
same thing, for the line-pairs which touch a given curve (m, n), the parametric point is 
here situate on a twofold locus, the intersection of the discriminant-locus with the con 
tact-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 conics Avhich 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 con 
sideration 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) q = 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 + 2 m). 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.