466 A MEMOIR ON ABSTRACT GEOMETRY. [413 
52. A onefold relation between the coordinates is expressible by means of an 
equation of the form 
(*$>> ^•••) ( ' ) = o. 
53. The expression “ an equation ” used without explanation may be taken to mean 
an equation of the form in question, viz. the equation obtained by putting a quantic 
equal to zero; the quantic is said to be the nil factum of the equation. We may 
consequently say simply that a onefold relation between the coordinates is always 
expressible by an equation. 
54. It is frequently convenient to denote the quantic or nilfactum by a single 
letter, and to use a locution such as “the equation TJ =(*]£#, y..,) (-) =0,” which really 
means that the single letter TJ stands for the quantic (*]£#, y,...) (,) , so that we are 
afterwards at liberty to write TJ = 0 as an abbreviated expression for (*$#, y, ...)"• = 0. 
We may also speak of the equation or function TJ = 0, meaning thereby the equation 
U —0, or the function TJ. 
55. A &-fold relation between the coordinates is (as has been shown) equivalent 
to a system of k or more onefold relations; each of these is expressible by an equation 
U=0, and the &-fold relation is thus expressible by a system of k or more such 
equations. Representing by ((TJ)) the system of functions which are the nilfacta of 
these equations respectively, the ¿-fold relations may be represented thus, ((U)) = 0 ; or 
more completely, the relation being &-fold, and the number of equations being =$, by 
the notation 
((U) s) (k-fold) = 0. 
We may also speak of the system or relation ((TJ)) = 0, meaning thereby the system of 
functions ((U)) y or the relation ((TJ)) = 0. 
Article Nos. 56 to 62. Resultant, Discriminant, &c. 
56. In the case k >m, a given &-fold relation between the m+ 1 coordinates 
(x, y,...) and the parameters (x\ y',...) leads to a (k — m)fold relation between the 
parameters. This is termed the resultant relation of the given ¿-fold relation, or when 
the additional specification is necessary, the resultant relation obtained by elimination 
of the coordinates (x, y,...). 
57. Consider a ¿-fold relation between the m + 1 coordinates (x, y, ...) and the 
m’ +1 coordinates (x, y',...). If k ^ m, then, considering the (x, y,...) as coordinates 
and the (x', y', ...) as parameters, we have corresponding to the given relation a &-fold 
locus in the m-space; and so if kjf>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.