we have here six parameters (a, b, c, f, g, h), which are taken as the coordinates of a 
406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
197 
the order r which touch a given curve of the order m and class n, the order of the 
contact-locus is = n + (2r — 2) m. 
14. If, then, the curve touch a given curve, the parametric point is situate on the 
contact-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, or any higher order, or has two or more ordinary or other contacts 
with the same given curve, then if the total manifoldness be = k, the parametric point 
is situate on a &-fold locus, which is given as a singular locus of the proper kind on 
the onefold contact-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 ordinary 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 inter 
sections 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 
equation is ax+ by + cz = 0, and the parameters (a, b, 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 con 
tact-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, hjcc, y, zf = 0,