PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
61 
We may also speak of the system or relation ((U)) = 0, meaning thereby the system of 
functions ((U)), or the relation ((U)) = 0. 
Resultant, Discriminant, &c. Article Nos. 56 to 62. 
56. In the case k > m, a given /'-fold relation between the m+ 1 coordiilates (x, y , . . .) 
and the parameters ( x ', y ', . . .) leads to a (/' — m)fold relation between the parameters. 
This is termed the resultant relation of the given Z>fold relation, or when the additional 
specification is necessary, the resultant relation obtained by elimination of the coordi- 
nates (x, y, . . .). 
57. Consider a /.’-fold relation between the m- j-1 coordinates (x, y, . . .) and the 
m'-\- 1 coordinates (x 1 , y ' , . . .). If kj>m, then, considering the (x, y, . . .) as coordinates 
and the ( x ', y', . . .) as parameters, we have corresponding to the given relation a Z>fold 
locus in the m-space; and so if k^i>m', then, considering the (x ! , y’, . . .) as coordinates, 
but the (x, y, . . .) as parameters, we have corresponding to the given relation a /-fold 
locus in the m'-space. 
58. If k>m, but if the {k — m)fold resultant relation is satisfied, then the given /-fold 
relation becomes a m-fold linear relation between the coordinates (x, y, . . .), and is con- 
sequently satisfied by a single set of values of the coordinates. Hence, considering the 
given /-fold relation as implying the (/' — m)fold resultant relation, the /'-fold relation 
will represent a single point in the m-space, say, the common point. 
59. A m-fold relation, or the locus, or point-system thereby represented, may have a 
double or nodal point, viz. two of the points of the point-system may be coincident. 
More generally a /'-fold relation (k$>m), or the locus thereby represented, may have a. 
double or nodal point; for let the relation if less than m-fold be made m-fold by adjoin- 
ing to it a linear (m — Z')fold relation satisfied by the coordinates of the point in ques- 
tion but otherwise arbitrary, then, if the point in question be a double or nodal point 
of the m-fold relation, or of the point-system thereby represented, the point is said to be 
a double or nodal point of the original Z>fold relation, or of the locus thereby repre- 
sented. 
60. A given /’-fold relation (/:;£> m) between the m-f-1 coordinates, or the locus 
thereby represented, has not in general a nodal point. But if the relation involve the 
m'-j-l parameters (x 1 , y', . . .), then, if a certain onefold relation be satisfied between the 
parameters, there will be a nodal point. The onefold relation between the parameters 
is the discriminant relation of the given /'-fold relation. 
61. In the case in question, k^>m, the discriminant relation is the resultant relation 
of a (m + ljfold relation which is the aggregate of the given /’-fold relation with a cer- 
tain relation called the Jacobian relation, or when the distinction is required, the Jaco- 
bian relation in regard to the (x, y , . . .). 
62. Consider a /Mold relation $>m!) between the m-f-1 coordinates (x, y, . . .) 
and the m' + l coordinates ( x ', y ', . . .). It has been seen that to a given set of values of 
