406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
193 
Article Nos. 1 to 23.—On the quasi-geometrical representation of Conditions. 
1. A condition imposed upon a subject gives rise to a relation between the 
parameters of the subject; for instance, the subject may be, as in the present Memoir, 
a plane curve of a given order, and the parameters be any arbitrary parameters con 
tained in the equation of the curve. The condition may be onefold, twofold,... or, 
generally, /¿-fold, and the corresponding relation is onefold, twofold, ... or /¿-fold accord 
ingly. Two or more conditions, each of a given manifoldness, may be regarded as 
forming together a single condition of a higher manifoldness, and the corresponding 
relations as forming a single relation; and thus, though it is often convenient to con 
sider two or more conditions or relations, this case is in fact included in that of a 
Wold condition or relation. In dealing with such a condition or relation it is assumed 
that the number of parameters is at least = k; for otherwise there would not in 
general be any subject satisfying the condition: when the number of parameters is 
= k, 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 ¿-fold condition or relation, and the same number k of parameters, the number 
of solutions is in general determinate. 
3. A condition may in some cases be 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 may be 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 corresponding 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 representation of a condition. 
4. A set of any number, say &>, of parameters may be considered as the coordi 
nates of a point in «-dimensional space; and if the parameters are connected by a 
onefold, twofold,... or Wold relation, then the point is situate on a onefold, twofold,... 
or Wold 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 «-fold, viz. it is in this 
case a point-system. The relation made up of a &-fold relation, an /-fold relation, &c., is 
in general (k + / -4- &c.) fold, and the corresponding locus is (k + l + &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 be attended to, so that the distinct points 
of the system may have to be reckoned each its proper number of times. The locus 
c. VI. 2 5