A MEMOIR ON ABSTRACT GEOMETRY. 
467 
413] 
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 consequently satisfied by a single set of values of the coordinates. Hence, considering 
the given /¿-fold relation as implying the (k — ?n)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 4> 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 
adjoining to it a linear (m — &)fold relation satisfied by the coordinates of the point 
in question 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 ¿-fold relation, or of the locus 
thereby represented. 
60. A given-fold relation (k }> m) between the m + 1 coordinates, or the locus 
thereby represented, has not in general a nodal point. But if the relation involve the 
m'-fl parameters (x\ 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 Z>fold relation. 
61. In the case in question, k rj> m, the discriminant relation is the resultant 
relation of a (m-f l)fold relation which is the aggregate of the given ¿-fold relation 
with a certain relation called the Jacobian relation, or when the distinction is required, 
the Jacobian relation in regard to the (x, y, ...). 
62. Consider a &-fold relation (k m, }> m') between the m + 1 coordinates (x, y, ...) 
and the m'-fl coordinates (x', y',...). It has been seen that to a given set of values of 
the (#', 2/',...) or, say, to a given point in the m'-space, there corresponds a &-fold locus 
in the m-space, and that to a given set of values of the (x, y,...), or to a given 
point in the m-space, there corresponds a &-fold locus in the m'-space. The &-fold 
locus in the m'-space may have a nodal point; this will be the case if there is 
satisfied between the (x, y, ...) a certain one-fold relation, the discriminant relation of 
the given ¿-fold relation in regard to the (x, y, ...). This onefold relation represents 
in the m-space a onefold locus, the envelope of the &-fold loci in the m-space corre 
sponding to the several points of the m'-space. The property of the envelope is that 
to each point thereof there corresponds in the m'-space a Z>fold locus having a nodal 
point. 
Article Nos. 63—69. Consecutive Points; Tangent Omals. 
63. As the notions of proximity and remoteness have been thus far altogether 
ignored, it seems necessary to make the following 
Postulate. We may conceive a point consecutive (or indefinitely near) to a given 
point. 
59—2