WHICH SATISFY GIVEN CONDITIONS. 
81 
14. If, then, the curve touch a given curve, the parametric point is situate on the con- 
tact-locus of that curve. If it touch a second given curve, the parametric point is in like 
manner situate on the contact-locus of the second given curve, that is, it is situate on 
the twofold locus which is the intersection of the two contact-loci ; and the like in the 
case of any number of contacts each with a distinct given curve. But if the curve, 
instead of ordinary contacts with distinct given curves, has either a contact of the second, 
or third, any higher order, or has two or more ordinary or other contacts with the same 
given curve, then if the total manifoldness be =Jc, the parametric point is situate on a 
Mold locus, which is given as a singular locus of the proper kind on the onefold con- 
tact-locus ; so that the theory of the contact-locus corresponding to the case of a single 
contact with a given curve, contains in itself the theory of any system whatever of ordi- 
nary or other contacts with the same given curve, viz. the last-mentioned general case 
depends on the discussion of the singular loci which lie on the contact-locus. And 
similarly, if the curve has any number of ordinary or other contacts with each of two or 
more given curves, we have here to consider the intersections of singular loci lying on 
the contact-loci which correspond to the several given curves respectively, or, what is 
the same thing, to the singular loci on the intersection of these contact-loci ; that is, the 
theory depends on that of the contact-loci which belong to the given curves respectively. 
15. Suppose that the curve which has to satisfy given conditions is a line ; the equa- 
tion is ax-\-by-\-cz= 0, and the parameters ( a , 5, c ) are to be taken as the coordinates of 
a point in a plane. Any onefold condition imposed upon the line establishes a onefold 
relation between the coordinates (a, b , c), and the parametric point is situate on a curve ; 
a second onefold condition imposed on the line establishes a second onefold relation 
between the coordinates (a, b, c), and the parametric point is thus situate on a second 
curve ; it is therefore determined as a point of intersection of two ascertained curves. 
In particular if the condition imposed on the line is that it shall touch a given curve, the 
locus of the parametric point is a curve, the contact-locus ; (this is in fact the ordinary 
theory of geometrical reciprocity, the locus in question being the reciprocal of the given 
curve ;) and the case of the twofold condition of a contact of the second order, or of 
two contacts, with the given curve, depends on the singular points of the contact-locus, 
or reciprocal of the given curve ; in fact according as the line has a contact of the 
second order, or has two contacts with the given curve (that is, as it is an inflexion- 
tangent, or a double tangent of the given curve), the parametric point is a cusp or a node 
on its locus, the reciprocal curve : this is of course a fundamental notion in the theory 
of reciprocity, and it is only noticed here in order to show the bearing of the remark 
(ante, No. 14) upon the case now in hand where the curve considered is a line. 
16. If the curve which has to satisfy given conditions is a conic 
(a, b, c, f, g , hjx, y , z) 2 =0, 
we have here six parameters (a, b, c,f, g , li), which are taken as the coordinates of a 
point in 5-dimensional space. It may be remarked that in this 5-dimensional space 
we have the onefold cubic locus abc — af 2 — bg' 1 — ch' 2 -{-2fgk—0, which is such that to 
