[417 
418] 
5 
as (exclusively of 
it the intersections 
of circles intersect 
i mentioned above, 
hree other points, 
418. 
nversation that he 
r the locus of the 
A REMARK ON DIFFERENTIAL EQUATIONS. 
[From the Philosophical Magazine, vol. xxxn. (1866), pp. 379—381.] 
Consider a differential equation f(x, y, p) = 0, of the first order, but of the degree 
n, where f is a rational and integral function of (cc, y, p) not rationally decomposable 
into factors: the integral equation contains an arbitrary constant c, and represents 
therefore a system of curves, for any one of which curves the differential equation is 
satisfied: the differential equation is assumed to be such that the curves are algebraical 
curves. The curves in question may be considered as undecomposable curves ; in fact, if 
the curve U a V p W y ... =0 (composed of the undecomposable curves U = 0, F=0, W = 0,..) 
satisfies the differential equation, then either the curves U — 0, V = 0, W= 0,. . each 
satisfy the differential equation, and instead of the curve TJ a V^W y ... =0 we have 
thus the undecomposable curves U = 0, V = 0, W = 0,.. each satisfying the differential 
equation; or if any of these curves, for instance W = 0, &c., do not satisfy the differential 
equation, then W y , &c. are mere extraneous factors which may and ought to be rejected, 
and instead of the original curve U a VPW y ... = 0, we have the undecomposable curves 
U — 0, V = 0 satisfying the differential equation. Assuming, as above, the existence of 
an algebraical solution, this may always be expressed in the form $ (x, y, c) = 0, where 
0 is a rational and integral function of (x, y, c), of the degree n as regards the 
arbitrary constant c: this appears by the consideration that for given values (x 0> y 0 ) 
of (x, y) the differential equation and the integral equation must each of them give 
the same number of values of p. It is to be observed that <f> regarded as a function 
of (x, y, c) cannot be rationally decomposable into factors; for if the equation were 
cf) = ... = 0, d>, ffq &c. being each of them rational and integral functions of (x, y, c), 
then the differential equation would be satisfied by at least one of the equations 
d> = 0, T = 0, ... that is, by an equation of a degree less than n in the arbitrary 
constant c.