WHICH SATISFY GIVEN CONDITIONS. 
77 
single condition of a higher manifoldness, and the corresponding relations as forming a 
single relation ; and thus, though it is often convenient to consider two or more condi- 
tions of relations, this case is in fact included in that of a Mold condition or relation. 
In dealing with such a condition or relation it is assumed that the number of parameters 
is at least ~Jc\ for otherwise there would not in general be any subject satisfying the 
condition : when the number of parameters is —Jc, the number of subjects satisfying the 
condition is in general determinate. 
2. A subject which satisfies a given condition may for shortness be termed a solution 
of the condition ; and in like manner any set of values of the parameters satisfying the 
corresponding relation may be termed a solution of the relation. Thus for a Mold 
condition or relation, and the same number Jc of parameters, the number of solutions is 
in general determinate. 
3. A condition may in some cases he satisfied in more than a single way, and if a 
certain way be regarded as the ordinary and proper one, then the others are special or 
improper : the two epithets maybe used conjointly, or either of them separately, almost 
indifferently. For instance, the condition that a curve shall touch a given curve (have 
with it a two-pointic intersection) is satisfied if the curve have with the given curve a 
proper contact ; or if it have on the given curve a node or a cusp (or, more specially, if 
it be or comprise as part of itself two coincident curves) ; or if it pass through a node 
or a cusp of the given curve : the first is regarded as the ordinary and proper way of 
satisfying the condition ; the other two as special or improper ways ; and the correspond- 
ing solutions are ordinary and proper solutions, or special or improper ones accordingly. 
This will be further explained in speaking of the locus which serves for the representa- 
tion of a condition. 
4. A set of any number, say a, of parameters may be considered as the coordinates of 
a point in ^-dimensional space ; and if the parameters are connected by a onefold, two- 
fold, ... or Mold relation, then the point is situate on a onefold, twofold, ... or Mold 
locus accordingly ; to the relation made up of two or more relations corresponds the 
locus which is the intersection or common locus of the loci corresponding to the several 
component relations respectively. A locus is at most Mold, viz. it is in this case a point- 
system. The relation made up of a Mold relation, an 7-fold relation, &c., is in general 
(&-f-Z+&c.)fold, and the corresponding locus is (#+7+&c.)fold accordingly. 
5. The order of a point-system is equal to the number of the points thereof, where, 
of course, coincident points have to he attended to, so that the distinct points of the 
system may have to be reckoned each its proper number of times. The locus corre- 
sponding to any linear j-£ old relation between the coordinates is said to be a^-fold omal 
locus ; and if to any given Mold relation we join an arbitrary (<y — £)fold linear relation, 
that is, intersect the Mold locus by an arbitrary (a — #)fold omal locus, so as to obtain 
a point-system, the order of the Mold relation or locus is taken to be equal to the 
number of points of the point-system, that is, to the order of the point-system. And 
this being so, if a Mold relation, an 7-fold relation, &c. are completely independent, that 
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