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LI. A Remark on Differential Equations. 

 By Professor Cayley, F.R.S* 



CONSIDER a differential equation f{x, y,p) = 0, of the first 

 order, but of the degree n f where /is a rational and inte- 

 gral function of (x, y,p) not rationally decomposable into factors : 

 the integral equation contains an arbitrary constant c, and repre- 

 sents 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 indecomposable curves ; 

 in fact, if the curve U a Y p W y . . . =0 (composed of the undecom- 

 posable curves U = 0, V = 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 

 XJ a y^ 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 differ- 

 ential equation, then W Y , &c. are mere extraneous factors which 

 may and ought to be rejected, and instead of the original curve 

 U a V^W 7 . . . =0, we have the undecomposable curves U=0, 

 V = satisfying the differential equation. Assuming, as above, 

 the existence of an algebraical solution, this may always be ex- 

 pressed in the form <$>{x,y, c) =0, where cf> is a rational and in- 

 tegral function of (x, y, c), of the degree nas regards the arbi- 

 trary constant c : this appears by the consideration that for given 

 values (x , y ) 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 cf> regarded as a function of (x, y, c) 

 cannot be rationally decomposable into factors; for if the equa- 

 tion were (£> = <&*$?. . . =0, <I>, M/*, &c. being each of them ra- 

 tional and integral functions of [x, y, c), then the differential 

 equation would be satisfied by at least one of the equations <£> = 0, 

 ^ = 0, . . . that is, by an equation of a degree less than n in the 

 arbitrary constant c. 



But the equation cf>(x, y,c) = is not of necessity the equation 

 of an undecomposable curve, and the undecomposable curve which 

 constitutes the proper solution of the differential equation cannot 

 always be represented by an equation of the form in question. 

 For although cj> regarded as a function of (x, y, c) is not ration- 

 ally decomposable into factors, yet it may very well happen that 

 cp> regarded as a function of (x, y) is rationally decomposable 

 into factors (geometrically the sections by the planes z — c of the 

 undecomposable surface <fi(x, y, z) =0 may each of them be com- 



* Communicated by the Author. 



