444 Prof. Cayley on Differential Equations and Umbilici, 

 gral equation finally is 



const. = (fa +gy + ^ Q ) " l 



^Ooo+9y + a(bx + cy)+Vny'Xfa+gy^P{bx + cy)^^n) 1 ^ 



(j*+9y + v{bx + cy) + Vn) l - c > 



where the quantities a, fi, 7, A, B, C are given by 



(b + cv)(v*-l)+2(f+gv) = c(v-u)(v-/3)(v- 7 ), 



c{v' 2 -l) + 2gv A B C 



t(v— *)(v— fi)(v— 7) "" v—ot v—/3 v — y 



Consider the curve 



°=(fa+gy+<bx+cy)+\/ay-Xfx+gyi-^{bx+c I/ )+ x /c}y' 1 

 (fa +gy + y{b* + cy) + \/d ) 1_c > 



which corresponds to the value =0 of the constant. If, for 

 instance, 



fa+gy + ct(bx-\-cy)+\/n=Q, 



this equation gives 



{bx + cy){{bx + cy){u*-l) + 2(fa+gy) a }=0; 

 or say 



(A* + cy) (a 2 - 1 ) + 2(> +$y)« = 0. 



But we have 



(& + Ca )(a 2 ~l) + 2(/+#«)« = 0, 

 and the equation therefore is 



(bx + cy) (/+£*) - (/# +gy) [b + c«) = ; 

 that is, 



(c/~ i0)(y*- «#)=0; 

 or simply 



that is, the directions of the curve at the origin, or point x=0, 

 ?/ = 0,are given by the equations y—ux — Q, y—j3a: = 0, y—<yx=0. 

 This is right, since from the differential equation we obtain at 

 the origin 



(b + cp)(p*-l) + 2(f+gp)p=:c(p-«)(p-/3)(p- r )=0. 



V. 



The particular case of the equation 



y ( p 2 — 1) + 2mxp = 



