82 
PROFESSOE CATLET OjST THE CURVES 
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 
(be— f 2 =0, ca—cf— 0, ab—h 2 — 0, gli—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, 
(*Ja, b, c,f,g,h)= 0, 
(#'X«, b, c,f, g, h)= 0; 
the equations of the locus are satisfied by 
a:b: c:f:g: h=a? : /3 2 : y 2 : j3y : yu, : 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: 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 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 maybe noticed that this twofold locus is of the order 3(w-j-2m), but that 
it breaks up into a twofold locus of the order 3 n, which gives the proper solutions ; 
viz. the nodal conics 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 3 m 
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 ) q = 0 (to fix 
the ideas, if the given curve is of the order m and class n, then the order g of the contact- 
locus is =n J r 2m). 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 contact at each 
of the points of intersection of the given curve with the coincident line-pair regarded as 
a single line, that is, in the case of a given curve of the m-th order, m special contacts, 
the Bipoint-locus is a multiple curve on the corresponding contact-locus. 
* In framing the epithet Bipoint, the coincident line-pair is regarded as being really a point-pair : see post, 
ISTo. 30. 
