418] 
A REMARK OK DIFFERENTIAL EQUATIONS. 
7 
functions of the single parameter c, the point to which they belong is an arbitrary 
point on a certain curve or (k— l)fold locus C in the ^-dimensional space. And this 
curve must be such that to given values of {x, y) there shall correspond n points on the 
curve; that is, treating (x, y) as constants, the surface or onefold locus yfr(x, y, c 1} c 2 ...C] C )=0, 
and the curve or (1c — l)fold locus C, shall' meet in n points. The conclusion stated 
in the foregoing quasi-geometrical form is, that the solution of the differential equation 
may be exhibited in the form ijr(x, y, c lf Ca...c*) = 0; viz. yjr is a rational and integral 
function of (x, y, Ci, Co... Cf;), where (c 1; c 2 ... Ck) are the coordinates of an arbitrary or 
variable point on a curve or (k — l)fold locus G in a ^-dimensional space, which curve 
meets the surface or onefold locus {pc, y, c x , c 2 ...c*) in n points, and where \jr 
regarded as a function of (x, y) is not rationally decomposable into factors. 
Cambridge, October 13, 1866.