196 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
the common (u> —j) fold locus, and the difficulty is in general in the determination of 
the value of this reduction. 
11. In all that precedes, the number of the parameters has been taken to be o> ; 
but if the parameters are taken to be contained in the equation of the curve homo 
geneously, then the parameters before made use of are in fact the ratios of these 
homogeneous parameters; and using the term henceforward as referring to the homo 
geneous parameters, the numbers of the parameters will be = <w + 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 j-fold linear relation between the parameters. It 
follows that the number of the curves which satisfy a given &-fold condition, and besides 
pass through co — k given points, is equal to the order of the /r-fold relation, or of the 
corresponding &-fold locus; and thus if we define the order of the ¿-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 4-1 + &c.) fold condition made up of a &-fold condition, an ¿-fold condition, &c., is 
equal to the product of the orders of the component conditions; and in particular if 
k + 1 + &c. = o), then the order of the w-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 para 
metric 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 onefold 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 corresponds 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