78 
PEOFESSOE CAYLEY ON THE CTTEVES 
is, if they are not satisfied by values which satisfy a less than (/£+/+ &c.)fold relation, 
or, what is the same thing, if the /c-fold locus, the /-fold locus, &c. have no common 
less than (7r+Z+&c.)fold locus, then the relations make up together a (&-f-Z-j-&c.)fold 
relation, and the loci intersect in a (/£-b/-f-&c.)fold locus, the orders whereof are 
respectively equal to the product of the orders of the given relations or loci. In parti- 
cular if we have Jc-\-l-\-8zc.=u, 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 Afold relation, an /-fold relation, &c., if they were together equivalent to a less 
than (Jc + 7+ &c.)fold relation, would not he independent ; but the relations, assumed to 
be independent, may yet contain a less than (£-j-Z-j-&c.)fold relation, that is, they may 
be satisfied by the values which satisfy a certain less than (& + Z-|-&c.)fold relation (say 
the common relation), and exclusively of these, only by the values which satisfy a proper 
(/£ + /-}-&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 7>fold locus, the /-fold locus, &c. 
may have in common a less than (#-f-Z-f-&c.)fold locus, and besides intersect in a resi- 
dual (#+7+&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 com- 
plete independence.) In particular if #+Z-{-&c. = <y, the several loci may intersect, say 
in an (u—j) fold locus, and besides in a residual a/-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 co arbitrary para- 
meters : 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 
<y-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 7>fold condition, then we have a 
onefold, twofold, ... or /c-fold relation between the parameters, and the parametric 
point is situate on a onefold, twofold, ... or 7>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 <w-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. 
8. Considering the general case where the condition, and therefore also the locus, is 
T’-fold, it is to be observed that every solution whatever, and therefore each special solu- 
tion (if any), corresponds to some point on the 7>fold locus ; we may therefore have on 
