The collected mathematical papers of Arthur Cayley.

242

ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.

[406

beyond the limits within which the present formulae are applicable; but this I am not
in a position to enter upon. If the extended formulae were obtained, it would of
course be an interesting verification or application of them to deduce from them the
complete series of expressions (1::), (2..)... (1, 1, 1, 1, 1) for the number of the conies
which satisfy given conditions of contact with a given curve, and besides pass through
the requisite number of given points. It will be recollected that throughout these last
investigations, I have put -De Jonquieres' p = 0; that is, I have not considered the case
of the curves C' which (among the conditions satisfied by them) have with the curve
Um contacts of given orders at given points of the curve; it is probable that the
general formulae containing the number p admit of extensions and transformations
analogous to the formulae in which p is put = 0, but this is a question which I have
not considered.
93. The set of equations (a)= [a], (a, b)= [a] [b] + [a, b], &c., considered irrespectively
of the meaning of the symbols contained therein, gives rise to an analytical question
which is considered in Annex No. 7.
The question of the conics satisfying given conditions of contact is considered from
a different point of view in my Second Memoir above referred to.
Annex No. 1 (referred to in the notice of DE JONQUIERES' memoir of 1861).-On the
form of the equation of the curves of a series of given index.
To obtain the general form of the equation of the curves CO1 of a series of the
index -W, it is to be observed that the equation of any such curve is always included in
an equation of the order n in the coordinates, containing linearly and homogeneously
certain parameters a, b, c,..; this is universally the case, as we may, if we please, take
the parameters (a, b, c,..) to be the coefficients of the general equation of the order n;
but it is convenient to make use of any linear relations between these coefficients so
as to reduce as far as possible the number of the parameters. Assume that the
number of the parameters is = o+ 1, then in order that the curve should form a
series (that is, satisfy In( n+3)-I conditions), we must have a ( - 1) fold relation
between the parameters, or, what is the same thing, taking the parameters to be the
coordinates of a point in co-dimensional space, say the parametric point, the point in
question must be situate on a (w - 1)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 (co-1)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 (w - 1) fold locus; or the index of the series being assumed to be = N,
the order of the ( - 1) fold locus must be also = N. That is, the general form of the
equation of the curves (7 which form a series of the index N, is that of an equation