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A MEMOIR ON ABSTRACT GEOMETRY. 
[413 
64. If the coordinates of the given point are (x, y,...), those of the consecutive 
point may be assumed to be (x+Bx, y + By,where Bx, By,... are indefinitely small 
in regard to (x, y, ...). 
65. It may be remarked that, taking the coordinates to be (x+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 (x + X y+Y,...) 
will be the very same with the original point, and it is therefore clear that a con 
secutive 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+Bx : y+By,..., where Bx, By,... are indefinitely small in regard to (x, y,...), will 
represent any set of ratios indefinitely near to the ratios (x : y,-..). 
The foregoing quantities (Bx, By, ...) are termed the increments. 
66. Consider a &-fold relation between the m + 1 coordinates (x, y,...), k^>m; the 
increments (Bx, By,...) are connected by a linear &-fold relation. 
The linear &-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 
(Bx, By,...)=(x, y, ...), since in such an equation only the ratios are attended to. But 
it may be preferable to write (Bx, By,...) = \(x, y,...). In particular if k=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 
proportional 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 &-fold 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 &-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 
meters, the locus of the point (x, y,...) is a &-fold ornai locus: it is to be observed 
that, by what precedes, the linear &-fold relation is satisfied by writing therein the 
values x : y,... =x : y,..., that is, the &-fold ornai locus passes through the original 
point (x, y,...); the &-fold ornai locus is said to be the tangent-omal of the original 
¿-fold locus at the point (x, y,...), which point is said to be the point of contact. 
68. If in the original ¿-fold locus we replace (x, y,...) by (x, y, ...), and combine 
therewith the &-fold linear relation, we have between the coordinates (x, y, ...) a 2&-fold 
relation (containing as parameters the coordinates (x, y,...)) ; these parameters satisfy 
the original &-fold relation, and in virtue hereof the 2&-fold relation (whether 2k is 
or is not greater than m) is satisfied by the values x, y,... =x : y :... ; and not only 
so, but the point in question is a nodal or double point on the 2&-fold locus. It 
also follows that the tangent-omal locus, considering in the ¿-fold linear relation 
(x, y,...) as parameters satisfying the original /¿-fold relation, has for its envelope the 
&-fold locus.