WHICH SATISFY GIVEN CONDITIONS. 
125 
series (that is, satisfy 3) — 1 conditions), we must have a [a— l)fold relation 
between the parameters, or, what is the same thing, taking the parameters to be the 
coordinates of a point in ^-dimensional space, say the parametric point, the point in 
question must be situate on a [a— l)fold locus. Moreover, the condition that the curve 
shall pass through a given point establishes between the parameters a linear relation 
(viz. that expressed by the original equation of the curve regarding the coordinates 
therein as belonging to the given point, and therefore as constants) ; that is, when the 
curve passes through a given point, the corresponding positions of the parametric point 
are given as the intersections of the (a — l)fold locus by an omal onefold locus; the 
number of the curves is therefore equal to the number of these intersections, that is, to 
the order of the (a>— l)fold locus; or the index of the series being assumed to be =N, 
the order of the (&>— l)fold locus must be also =N. That is, the general form of the 
equation of the curves C n which form a series of the index N, is that of an equation of 
the order n containing linearly and homogeneously the u -\- 1 coordinates of a certain 
(oo — l)fold locus of the order N. It is only in a particular case, viz. that in which the 
(a— l)fold locus is unicursal, that the coordinates of a point of this locus can be ex- 
pressed as rational and integral functions of the order N of a variable parameter 0 ; and 
consequently only in this same case that the equation of the curves C“ of the series of 
the index N can be expressed by an equation (#X^ ? V-> z) n =0, or y , l) n =0, 
rational and integral of the degree N in regard to a variable parameter 0. 
If in the general case we regard the coordinates of the parametric point as irrational 
functions of a variable parameter 0, then rationalising in regard to 0, we obtain an equa- 
tion rational of the order N in 0 , but the order in the coordinates instead of being =n, 
is equal to a multiple of n , say qn. Such an equation represents not a single curve but 
q distinct curves C”, and it is to be observed that if we determine the parameter by sub- 
stituting therein for the coordinates their values at a given point, then to each of the N 
values of the parameter there corresponds a system of q curves, only one of which 
passes through the given point, the other q — 1 curves are curves not passing through 
the given point, and having no proper connexion with the curves which satisfy this con- 
dition. 
Returning to the proper representation of the series by means of an equation con- 
taining the coordinates of the parametric point, say an equation (*Xx, V-> 1)“=0, in- 
volving the two coordinates (x, y ), it is to be noticed that forming the derived equation 
and eliminating the coordinates of the parametric point, we obtain an equation rational 
in the coordinates (x, y), and also rational of the degree N in the differential coefficient 
; in fact since the number of curves through any given point (# 0 , y 0 ) is =N, the 
differential equation must give this number of directi6ns of passage from the point 
{x 0 , y 0 ) to a consecutive point, that is, it must give this number of values of jr, and must 
consequently be of the order N in this quantity. 
