380 Prof. Cayley on Differential Equations. 



posed of two or more distinct curves) ; and assuming that the 

 function <£ is thus decomposed into its prime factors, then each 

 factor equated to gives an undecomposable curve satisfying the 

 differential equation, and constituting the proper solution thereof. 

 It may be observed that, by the foregoing process of decom- 

 position, we sometimes reduce the original equation <£(<#, y, c) =0 

 into a like equation ^(a?, y, c{) = of a more simple form. Thus, 

 for instance, if we have <j>{x, y, c) = U 2 — c = 0, U being a rational 

 and integral function of (pc, y), then instead of (£ = TJ 2 — c=0 we 

 have the equations U+ Vc = 0, U— v/c=0, each of which is an 

 equation of the form U— Cj = 0, or we pass from the original 

 equation <f>(w, y, c) = U 2 — - c = to the simplified equation 



cf) l (w,y f c i )=V — c ] =0. 



Or, to take a somewhat more complicated instance, if the given 

 integral equation be 



<f>(x,y 3 c) = U 4 + c 2 V 4 + (c+l) 2 W 4 



_2cU 2 V 2 -2(c + l)U 2 W 2 -2c(c + l)V 2 W 2 =0, 



then the equation U + V Vc + W\/c-\- 1=0, writing therein 



\/c= 2 * , and therefore Vc + 1 = — \ — =-, becomes 

 Cj *-~ 1 Cj —■ J- 



U( Cl 2 -l)+V.2^ + W( Cl 2 + l)=0; 



so that we pass from the original equation $>{oc,y } c)=0 to the 

 simplified equation 



0(^,y,c 1 )=U(c 1 2 -l) + V.2 Cl + W(c 1 2 + l) = O. 



But observe that the possibility of the rationalization depends on 

 the form of the radicals Vc and V'c+l ; if we had had Vc 

 and ^c 2 + l (or c and vV-fl), the rationalization could not 

 have been effected. 



Returning to the case of an integral equation <j>(w } y, c)=0, 

 where <j> regarded as a function of (#, y) is decomposable into fac- 

 tors, then equating to zero any one of the prime factors of (f>, we 

 obtain an integral equation ^{x } y, c X) c 2 . . . c k ) = 0, where 

 c v c 2 . . Cjc are irrational functions (not of necessity representable 

 by radicals, and without any superior limit to the number of these 

 functions) of c : here i/r regarded as a function of (#, y) is of 

 course undecomposable, and the equation yfrfa, y, c,, c 2 . . . c k ) =0 

 belongs to the undecomposable curve which is the proper solution 

 of the differential equation. The result may be stated under a 

 quasi- geometrical form ; viz. regarding c v c 2 . . . c k as the coor- 

 dinates of a point in /^-dimensional space, then as these are func- 

 tions 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 



