The collected mathematical papers of Arthur Cayley.

406]

ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.

243

of the order n containing linearly and homogeneously the w + 1 coordinates of a certain
( - 1) fold locus of the order N. It is only in a particular case, viz. that in which
the (w - 1) fold locus is unicursal, that the coordinates of a point of this locus can
be expressed 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
CG of the series of the index N can be expressed by an equation (.*x, y, z)' = 0,
or (G*x, y, 1)- = 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 equation 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 Cn, and it is to be observed that if we
determine the parameter by substituting 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 condition.
Returning to the proper representation of the series by means of an equation containing the coordinates of the parametric point, say an equation (G*x, y, 1) = 0,.
involving 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
dy; in fact since the number of curves through any given point (x0, y,) is =N, the
differential equation must give this number of directions of passage from the point
(x,, yo) to a consecutive point, that is, it must give this number of values of y, and
dx'
must consequently be of the order N in this quantity.
Conversely, if a given differential equation rational in x, y, -, and of the degree
N in the last-mentioned quantity d-, admit of an algebraical general integral, the
curves represented by this integral equation may be taken to be irreducible curves,.
and this being so they will be curves of a certain order n forming a series of the
index N; whence the general integral (assumed to be algebraical) is given by an equation
of the above-mentioned form, viz. an equation rational of a certain order n in the
coordinates, and containing linearly and homogeneously the o+1 coordinates of a
variable parametric point situate on an (a -1) fold locus. The integral equation
expressed in the more usual form of an equation rational of the order N in regard to
the parameter or constant of integration, will be in regard to the coordinates of an
order equal to a multiple of n, say = qn, and for any given value of the parameter
will represent not a single curve Cn, but a system of q such curves: the firstmentioned form is, it is clear, the one to be preferred.
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