350 On the Sextactic Points of a Plane Curve. [June 



Q = (vy-wft) 9* + (wauy) d y + (uft-vn) 9*, 

 then, writing as usual 



A= 1 Wj w' 2 , . . F=v' M?' u l u', . . 

 the values of the ratios a : b : c :/: g : h are determined by the equations 



v=o, n v =. Q 4 v=o. 



Now, if at the point in question the curvature of U be such that a sixth 

 consecutive point )ies on the conic V, the point is called a sextactic point ; 

 and the condition for this will be, in terms of the above formulae, Q 5 V=0. 

 From the six equations V=0, QV=0,... Q 5 V=0, the quantities a, b,.. h 

 can be linearly eliminated, and the result will be an equation which, when 

 combined with U = 0, will determine the ratios of x : y : z, the coordinates 

 of the sextactic points of U. But the equation so derived contains (beside 

 other extraneous factors) the indeterminate quantities a, ft, y, to the degree 

 15, which remain to be eliminated. Instead, however, of proceeding as 

 above, I eliminate a, /3, y beforehand, in such a way that V=0, Q V=0, 

 =0 take the form 



M w tzti. 



and more generally if W=0, representing any one of the series V=0, 

 n V=0, . . from which a, ft, y have been already eliminated, the equa- 

 tions W=0, QW=0, n 2 W=0 are replaced by 



ds W_d y W_d z W_ AW 

 u v w wil' 



where H is the Hessian of U, or a numerical factor, and 



Proceeding in this way, I obtain a result free from a, ft, y in the three 

 forms, 



=0, 



d z (wX a-P) d 2 (Y yP) 

 A( M X-.rP) A(wY-yP) 



9 x (X-xQ) d,(>Y-7/Q) 



where 



