38 W. Spottiswoode. Geometrical Theorems. [May 25, 



Substituting in equation (A), we obtain — 



D 3 ?/, xVhj + 3mj, 2yDhj + SDyDhj =0, 

 Bhj, xB 4 y + 4J)% 2yT)hj + %Vy'D s y + 6(D 2 y) 2 , 

 D 5 ?/, dD 5 7/ + 5D 4 ?/, 22/D 5 7/ + 10Di/D 4 2/ + 20D 2 ?/D 3 ?/. 



Whence by obvious inductions — 



Dhj, SDhj, =0 (B). 



Wy, AB% 3T>% 

 Dhj, 5D 4 #, lODhj. 



Also D{4(D 3 t/) 2 -3D 2 ?/ . ~Dhj}=4£)hj . T)*y-SDhj . T)*y. 



Hence the equation (B) may also be written thus — 



(10D 3 ^/-3D 2 i/ .D){4(D 3 2 /) 2 -3D 2 i /.D^}=0 . . (0). 



And in order to evaluate the expression it will consequently be 

 necessary, in the first instance at least, to calculate only Dy, . . . D 4 y. 



The main interest in the problem, however, lies in the determination 

 of the degree of this equation, which has been found by Cayley and 

 myself to be 12% — 27. The difficulty lies in getting rid of the ex- 

 traneous factors. In order to effect this, the expression (C) must be 

 transformed by the substitution of — u-.v for ~Dy, and other expressions 

 to be calculated for D 2 y, . . . D 4 ?/. 



If U=0 be the equation to the curve, and if we adopt a usual 

 notation, and put u, v for the first, and u lt w f , v± for the second differ- 

 ential coefficients of V with respect to x and y ; also if 



(v^x — uSy) ' 2 U = vhi-L — 2VUW ' + IV^, 



and similarly for higher powers, i.e., if the operative factor (vc x —uby)' 

 is understood to affect U only, precisely as if u, v, &c, were constant ; 

 then it is not difficult to show that — 



— Dy =u : v, 



— Dhj=(vc x —u(i !/ y 5 TJ : v i —3(vw , — wv l )(vd x —uc ]/ ) / ' 2 'U : v 5 , 



— J)±y=(vS x — wOyY^U : v'°— 4<(vw f — wv{)(v<5 x — ud^^JJ : v & 



— 3 { 2v (vh x — uSy)' 2 — 5 (W — uv{) 2 — v 2 (u l v 1 — iv' 2 ) } (vc x — uB^yjJ :v7. 



These are the developments by means of which the reduction would 

 have to be made. 



