406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
243 
31—2 
of the order n containing linearly and homogeneously the g> + 1 coordinates of a certain 
(ft) 1) fold locus of the order IV. 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 IV of a variable para 
meter 0; and consequently only in this same case that the equation of the curves 
C n of the series of the index IV can be expressed by an equation (*]£#, y, z) 11 — 0, 
or (*\x, y, l) n = 0, rational and integral of the degree IV in regard to a variable 
parameter 0. 
If in the general case we regard the coordinates of the parametric point as 
irratioual functions of a variable parameter 9, then rationalising in regard to 9, we 
obtain an equation rational of the order IV in 9, 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 n , 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 IV 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 con 
taining the coordinates of the parametric point, say an equation (*][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 IV in the differential coefficient 
^ ; in fact since the number of curves through any given point (x 0 , y n ) is = IV, the 
differential equation must give this number of directions of passage from the point 
(# 0 , y 0 ) to a consecutive point, that is, it must give this number of values of and 
must consequently be of the order IV in this quantity. 
Conversely, if a given differential equation rational in x, y, and of the degree 
dy 
dx’ 
admit of an algebraical general integral, the 
IV in the last-mentioned quantity > 
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 co +1 coordinates of a 
variable parametric point situate on an (co — 1) fold locus. The integral equation 
expressed in the more usual form of an equation rational of the order IV 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 C n , but a system of q such curves: the first- 
mentioned form is, it is clear, the one to be preferred.