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ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
corresponding to any linear y-fold relation between the coordinates is said to be a J-fold 
omal locus; and if to any given /r-fold relation we join an arbitrary (co — Jc) fold linear 
relation, that is, intersect the ¿-fold locus by an arbitrary (« — Jc) fold omal locus, so as 
to obtain a point-system, the order of the &-fold 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 &-fold relation, an ¿-fold relation, &c. are completely 
independent, that is, if they are not satisfied by values which satisfy a less than 
(Jc + l + &c.) fold relation, or, what is the same thing, if the /r-fold locus, the ¿-fold 
locus, &c., have no common less than (Jc +1 + &c.) fold locus, then the relations make 
up together a (Jc +1 + &c.) fold relation, and the loci intersect in a (Jc + l+ &c.) fold 
locus, the orders whereof are respectively equal to the product of the orders of the 
given relations or loci. In particular if we have Jc +1 + &c. = co, then we have an 
«-fold relation, and corresponding thereto a point-system, the orders whereof are 
respectively equal to the product of the orders of the given relations or loci. 
6. A &-fold relation, an ¿-fold relation, &c., if they were together equivalent to a 
less than (Jc + A + &c.) fold relation, would not be independent; but the relations, assumed 
to be independent, may yet contain a less than (Jc + ¿ + &c.) fold relation, that is, they 
may be satisfied by the values which satisfy a certain less than (Jc + l + &c.) fold relation 
(say the common relation), and exclusively of these, only by the values which satisfy 
a proper (Jc + ¿ + &c.) fold relation, which is, so to speak, a residual equivalent of the 
given relations. This is more clearly seen in regard to the loci; the /r-fold locus, the 
¿-fold locus, &c. may have in common a less than (Jc + l + &c.) fold locus, and besides 
intersect in a residual + ¿ + &c.) fold locus. (It is hardly necessary to remark that 
such a connexion between the relations is precisely what is excluded by the foregoing 
definition of complete independence.) In particular if Jc +1+ &c. =«, the several loci 
may intersect, say in an (co —j) fold locus, and besides in a residual «-fold locus, or 
point-system. The order (in any such case) of the residual relation or locus is equal 
to the product of the orders of the given relations or loci, less a reduction depending 
on the nature of the common relation or locus, the determination of the value of 
which reduction is often a complex and difficult problem. 
7. Imagine a curve of given order, the equation of which contains « arbitrary 
parameters: to fix the ideas, it may be assumed that these enter into the equation 
rationally, so that the values of the parameters being given, the curve is uniquely 
determined. Suppose, as above, that the parameters are taken to be the coordinates 
of a point in «-dimensional space; so long as the curve is not subjected to any 
condition, the point in question, say the parametric point, is an arbitrary point in the 
«-dimensional space; but if the curve be subjected to a onefold, twofold,... or A>fold 
condition, then we have a onefold, twofold,... or A;-fold relation between the parameters, 
and the parametric point is situate on a onefold, twofold,... or ¿b-fold locus accordingly: 
to each position of the parametric point on the locus there corresponds a curve 
satisfying the condition, that is, a solution of the condition. In the case where the 
condition is «-fold, the locus is a point-system, and corresponding to each point of 
the point-system we have a solution of the condition; the number of solutions is 
equal to the number of points of the point-system.