VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 517 



double edges of K. This triple point is thus equivalent to three actual 

 double points. If, however, when we deform K we also deform M in 

 such a way that it has a double point at the double point of Cm, it is 

 evident that the three tangents in the limit will not lie in one plane. 

 For they are the intersections of the tangent cone at the double point of 

 M by the three tangent planes to K along the triple line. The tangent 

 line to the third or new branch intersects the plane of the tangents to 

 the other two branches in the limit at an angle ; the new branch must 

 therefore be considered as meeting the first two branches together once 

 at that point. The two double edges of K caused by the third sheet 

 cannot in this case both be considered as due to actual double points of 

 Cm , as the three tangents would then in the limit lie in one plane. Nor 

 can both double edges be considered as due to apparent double points, 

 for the new branch actually meets the other two branches together in the 

 limit. One must therefore be considered as due to an actual and one to 

 an apparent double point. This kind of triple point is thus equivalent to 

 two actual and oue apparent double point. This agrees with what we 

 obtained on page 512. 



II. Quadruple Points. 



(1). If the four tangents all lie in one plane, as in 1,* the quadruple 

 point is equivalent to 6 actual double points. This is shown in the same 

 way as the case of the triple point of the first kind. 



(2). If three tangents lie in one plane, as in 2* the quadruple point 

 is equivalent to 4 actual and 2 apparent double points. For if a curve 

 having such a quadruple point is obtained as the partial intersection of a 

 cone A" and a monoid M, the point must be at least a triple point and the 

 line from it to the vertex at least a double Hne of kind III on M\ other- 

 wise three tangents cannot lie in one plane. Let a curve Cm with a 

 triple point at which the tangents lie in one plane be obtained as the 

 partial intersection of a cone K and a monoid M that has the point as 

 a triple point. This means that the three lines in which (in addition to 

 the line from the multiple point to the vertex 6 times) the three tangent 

 planes to K intersect the tangent cone at the triple point of M lie in one 

 plane. Let a new branch of Cm passing near the triple point be caused 

 by another sheet of K intersecting M. This sheet meets the three sheets 

 through the triple edge of K in three lines, which are thus double lines 

 on K. These lines must be due to actual or apparent double points 



* These numbers and succeeding ones refer to figures on the accompanying 

 plate. 



