516 PROCEEDINGS OF THE AMERICAN ACADEMY. 



also, 



P a+k + Pa-h-\ + g (g + 2 k + 1) apparent double points 



= P a+k+ g- 



a-\-k-\-g T P a — k— g— 1 • 



A sextic curve can thus have a quadruple point and four apparent 

 double points, or a triple point, a double point, and six apparent double 

 points. A quadruple point on a sextic curve is thus equivalent to a 

 triple point, a double point, and two apparent double points. 



Similarly a septimic curve can have a quintuple point and five appar- 

 ent double points, or a quadruple point, a double point, and eight apparent 

 double points, or two triple points and nine apparent double points. A 

 quintuple point on a septimic is thus equivalent to a quadruple point, a 

 double point, and three apparent double points, or to two triple points 

 and four apparent double points. Likewise a quadruple point and a 

 double point on a septimic curve are equivalent to two triple points and 

 one apparent double point. 



2. We shall now examine different kinds of multiple points and deter- 

 mine the number of actual and apparent double points to which they are 

 equivalent. We shall consider in detail all kinds of multiple points from 

 the triple point to the septuple point inclusive, and shall then draw some 

 conclusions for multiple points in general. 



I. Triple Points. 



A triple point on a curve can, as we have seen, be of two kinds 

 according as the tangents to the curve at this point do or do not lie in 

 one plane. We shall consider the way in which a triple point can be 

 formed. A curve C m with a double point can be obtained as the partial 

 intersection of a cone K and a monoid 31, where the double point of C m 

 is a double point on A' and an ordinary point of M. The line from this 

 point to the common vertex of A' and M is a double edge on A" but does 

 not lie on M. Let another sheet of A that passes near this line intersect 

 M in a new branch of C m . This new sheet of K intersects the two 

 sheets through the double line in two lines that are also double edges 

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

 points of C m . If A is deformed in such a way that these two edges 

 coincide with the first double edge and form a triple edge, C m will have 

 a triple point. The tangents at this triple point will all lie in one plane, 

 as they are cut out of the three tangent planes to K along the triple line 

 by the one tangent plane to 71/ at this point. The last two double edges 

 of K must have been due to actual double points of C m , these double 

 points being the points of intersection of the ordinary sheet of M by the 



