518 PROCEEDINGS OF THE AMERICAN ACADEMY. 



of C m . As in the case of the triple point it is evident that one and no 

 more must in the limit be considered as due to an actual double point. 

 This quadruple point is thus equivalent to the triple point, one actual 

 double point, and two apparent double points, that is, to four actual and 

 two apparent double points. 



(3). If no three tangents lie in one plane, as in 3, the quadruple point 

 is equivalent to 3 actual and 3 apparent double points. For, a curve 

 having a triple point at which the tangents lie on a quadric cone but not 

 in a plane can be obtained as the partial intersection of a cone K and a 

 monoid M that has the point as a double point and a line from it to the 

 vertex as an ordinary line of kind III. A new branch of the curve can 

 be added, as in the previous cases, to form a quadruple point. The tan- 

 gent to this branch will be cut out of the tangent cone at the double point 

 of M by the tangent plane to the new sheet of K. As this tangent does 

 not in general lie in the plane of any two other tangents, the new branch 

 must be considered as meeting the three branches together once at the 

 triple point. This quadruple point is therefore equivalent to the triple 

 point and one actual and two apparent double points, that is, to three 

 actual and three apparent double points. This agrees with what we 

 obtained on pages 501 and 516. 



We can go on in this way obtaining the different kinds of points of any 

 multiplicity by adding a new branch to points of one less multiplicity. 



III. Quintuple Points. 



(1). If all the tangents lie in one plane, as in 4, the quintuple point 

 is equivalent to 10 actual double points. This is obtained by adding 

 a new branch to a II (1) * in such a way that its tangent lies in the 

 plane of the four tangent lines at the quadruple point. 



(2). If four tangents lie in one plane, as in 5, the quintuple point is 

 equivalent to 7 actual and 3 apparent double points. This can be 

 obtained from a II (1). 



(3). If three tangents lie in one plane, and the other two lie in a plane 

 with one of these, as in 6, the quintuple point is equivalent to 6 actual 

 and 4 apparent double points. This can be obtained from a II (2) by 

 adding a new branch in such a way that its tangeut in the limit lies in a 

 plane with the tangents to two other branches. It is therefore equiva- 

 lent to a II (2), two actual double points and two apparent double points. 

 In Figure 6, let us consider the line 123 as the cross section of a plane 

 that contains the lines whose bases are 1, 2, and 3, and the line 145 as 



* We mean by II (1) a quadruple point of kind (1), etc. 



