VAN DER VRIE3. — MULTIPLE POINTS OF TWISTED CURVES. 525 



just three apparent double points when two actual and one apparent 

 double points are taken to form the triple point. The curve can be 

 obtained from a') or a") only by having one or two apparent double 

 points change to actual double points in the limit; this produces no new 

 kind of curve. The curve cannot be obtained from b), b'), or c) as there 

 will not be enough apparent double points remaining. There is thus 

 only one species of quintic curve having a triple point, and it has three 

 apparent double points. 



B. 



Sextic Curves, 

 A sextic curve, according to the classification of Salmon,* can have 



a) 4,6; a') 3,7; a") 2,8; a'") 1,9; a"^) 0,10; 



b) 3,6; b') 2,7; b") 1,8; b'") 0,9; 



c) 2,6; c') 1,7; c") 0,8; 



d) 1,6; d') 0,7; 



e) 0,6. 



In addition to these are the sextics that have multiple points. 



1. A sextic curve may have a quadruple point. The tangents at this 

 point cannot lie in one plane, as this plane would meet the curve in eight 

 points. A quadruple point at which three tangents lie in one plane is 

 equivalent, as we have seen, to four actual and two apparent double 

 points. A sextic with such a quadruple point thus seems to be derivable 

 from a curve of class a) above. The plane of the three tangents wouM, 

 however, meet the curve in seven points, which is not possible in the 

 case of a twisted sextic. It is thus evident that although a curve may 

 liave actual and apparent double points sufficient to form a certain 

 multiple point nevertheless there are cases where the points cannot 

 unite to form this multiple point. This is analogous to the case of plane 

 curves t The oidy quadruple point possible on a sextic curve is one at 

 which no three tangents lie in one plane. This curve has been con- 

 sidered on page 511, as the complete intersection of a quadric cone and a 

 cubic monoid. It has four apparent double points. The quadruple point 

 on this curve is equivalent to three actual and three apparent double 

 points. The curve can thus be obtained directly from a') above. It 

 cannot be formed from a) as it cannot have a double point in addition to 



* Salmon, Cambridge & Dublin Mathematical Journal, Vol. V. 

 t Salmon's Higher Plane Curves (1873), p. 27. 



