80 
PEOEESSOE CAYLEY ON THE CUEVES 
the parameters are taken to be contained in the equation of the curve homogeneously, then 
the parameters before made use of are in fact the ratios of these homogeneous para- 
meters ; and using the term henceforward as referring to the homogeneous parameters, 
the numbers of the parameters will be =w-f- 1. 
12. I assume also that the equation of the curve contains the parameters linearly: 
this being so, the condition that the curve shall pass through a given arbitrary point 
implies a linear relation between the parameters ; and the condition that the curve 
shall pass through j given points, a y'-fold linear relation between the parameters. It 
follows that the number of the curves which satisfy a given &-fold condition, and besides 
pass through a — k given points, is equal to the order of the #-fold relation, or of the 
corresponding #-fold locus ; and thus if we define the order of the k- fold condition to be 
the number of, the curves in 'question, the condition, relation, and locus will be all of the 
same order, and in all that precedes we may (in place of the order of the relation or of the 
locus) speak of the order of the condition. Thus, subject to the modifications occasioned 
by common loci and special solutions as above explained, the order of the ( k-\-l-\- &c.)- 
fold condition made up of a #-fold condition, a Z-fold condition, &c., is equal to the 
product of the orders of the component conditions; and in particular if k-\-l-\- &c. =a, 
then the order of the ay-fold condition, or number of the solutions thereof, is equal to 
the product of the orders of the component conditions. 
13. The conditions to be satisfied by the curve may be conditions of contact with a 
given curve or curves. In particular if the curve touch a given curve, the parametric point 
is then situate on a onefold locus. It is to be noticed in reference hereto that if the 
given curve have nodes or cusps, then we have special solutions, viz. if the sought for 
curve passes through a node or a cusp of the given curve ; and each such node or cusp gives 
rise to a special onefold locus, presenting itself in the first instance as a factor of the one- 
fold locus of the parametric point ; this is, however, a case where the special locus is of 
the same manifoldness as the general locus (ante, No. 8), and is consequently separable ; 
throwing off therefore all these special loci, we have a onefold locus which no longer 
comprises the points which correspond to curves passing through a node or a cusp of the 
given curve ; the onefold locus, so divested of the special onefold factors, may be termed 
the “ contact-locus ” of the given curve. To each point of the contact-locus there corre- 
sponds a curve having with the given curve a two-pointic intersection, viz. this is either a 
proper contact, or it is a special contact, consisting in that the sought for curve has on the 
given curve a node or cusp, or (which is a higher speciality) in that the sought for curve 
is or contains as part of itself two or more coincident curves (ante. No. 3). To a point 
in general on the contact-locus there corresponds a curve having a proper contact with 
the given curve, save and except that to each point on any one of certain special loci on 
the contact-locus there corresponds a curve having some kind of special contact as above 
with the given curve. To fix the ideas, it may be mentioned that for the curves of the 
order r which touch a given curve of the order m and class n, the order of the contact- 
locus is =w+(2r— 2)m. 
