62 
PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
the (x 1 , y', . . .) 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 4- fold locus in the m'-space. The yf-fold locus in the 
m'-space may have a nodal point ; this will be the case if there is satisfied between the 
( x , ?/,...) a certain onefold relation, the discriminant relation of the given &-fold relation 
in regard to the (x 1 , y',. . .). This onefold relation represents in the m-space a onefold 
locus, the envelope of the &-fold loci in the m-space corresponding to the several points 
of the m'-space. The property of the envelope is that to each point thereof there cor- 
responds in the m'-space a /r-fold locus having a nodal point. 
Consecutive Points ; Tangent Omals. Article Nos. 63-69. 
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. 
64. If the coordinates of the given point are ( x , y , . . .), those of the consecutive point 
may be assumed to be (x -f- cA, y-\-ly, . . .), where lx, ly , . . . are indefinitely small in regard 
to (x, y , . . .). 
65. It maybe remarked that, taking the coordinates to be (a’-j-X, y-\- Y, . . .), there is 
no obligation to have (X, Y, . . .) indefinitely small ; in fact whatever the magnitudes of 
these quantities are, if only X : Y : . . . =x : y : , then the point (a+X,y-(-Y, . . .) will 
be the very same with the original point, and it is therefore clear that a consecutive 
point may be represented in the same manner with magnitudes, however large, of X, Y, . . . 
But we may assume them indefinitely small, that is, the ratios x-\-lx : y-\-ly, • . ., where 
lx, ly, . . . are indefinitely small in regard to (x, y, . . .), will represent any set of ratios 
indefinitely near to the ratios (x : y, . . .). 
The foregoing quantities (lx, ly, . . .) are termed the increments. 
66. Consider a /Nfold relation between the m-j-T coordinates (x, y, . . .), Jc >> m ; the 
increments (<Lr, ly, . . .) are connected by a linear /r-fold relation. 
The linear Z;-fold relation is satisfied if we assume the increments proportional to the 
coordinates — this is, in fact, assuming that the point remains unaltered. We may write 
(lx, ly , . . . ) = (x, y , . . .), since in such an equation only the ratios are attended to. But it 
may be preferable to write (lx, ly , . . .)=\(x, y, . . .). In particular if Jc=m, then the 
increments are connected by a linear m-fold relation ; that is, the ratio of the increments 
is uniquely determined ; and as the relation is satisfied by taking the increments propor- 
tional to the coordinates, it is clear that the values which the linear m-fold relation gives 
for the increments are in fact proportional to the coordinates : viz. there is not in this 
case any consecutive point. 
67. Considering the yhfold relation as belonging to a #-fold locus in the m-space, so 
that (x, y, . . .) are the coordinates of a point on this locus, then if in the linear T’-fold 
relation between the increments these increments are replaced by the coordinates (x, y, . . .) 
of a point in the m-space, then considering the original coordinates (x, y , . . .) as para- 
