56 



Frederick C. Ferry. 



where o and x denote double points and cusps arising from 

 the intersection of branches lying- in the same sheet. If P' 

 lie on the curve cp = 0, this number N must be diminished 

 by two. 



Denoting by an inflexion a point where a curve is met 

 in three consecutive points by the tangent conic at the point, 

 we proceed to find the number of such points which we will 

 represent by i. As in plane curves, the inflexions are the 

 intersections of the curve cp = with the curve whose 

 equation is the determinant of the second derivatives of the 



given curve, i. e., 



H 



0^(D 



02<p 



o^cp 



oX^ op, oX ov 8À 



ô^çp o^cp Ô^Cp 



§2rp 



O^CO 



O^Cp 



SX 5v ou, ÔV ôv^ 



0, 



the Hessian of co = 0. 



The number of inflexions is accordingly the number of 

 intersections of cp = and H ^ , reduced by 6 for each 

 double point and 8 by each cusp (formed by branches lying 

 in the same sheet in that region). 



Hence 



