528 PROCEEDINGS OF THE AMERICAN ACADEMY. 



determined by 15 points In order to make it contain the curve we 

 must make it contain 19 points of it. As the vertex counts for two of 

 these points, it is only necessary to make the monoid contain 17 addi- 

 tional points of the curve. If we take the triple point of the curve as 

 one of the 15 points at our disposal, it is evident that there are just 

 enough points to make the monoid contain the curve. The quiutic cone 

 meets the inferior cone of the monoid in 10 lines, of which one is the 

 tangent to the curve at the vertex and the other nine are lines common 

 to the two surfaces. As there are only six lines on the monoid, it is 

 evident that at least three lines common to cone and monoid are double 

 lines on the cone due to apparent double points of the curve. The cone 

 can, however, not have more than three double edges in addition to the 

 triple edge ; every line of the monoid must therefore lie on the cone. 

 This sextic has therefore three apparent double points when viewed 

 from an ordinary point of the curve, or seven apparent double points 

 when viewed from an arbitrary point. 



We can obtain the three species directly from b), a), and a'), respect- 

 ively. As the triple point is equivalent to three actual double points, it 

 is evident that no other species can be obtained. The sextic that has a 

 double point in addition to this triple point can also be obtained as the 

 intersection of a quartic cone and a cubic monoid that have the double 

 point of the curve as their common vertex. 



C. 



Septimic Curves. 



A septimic curve, according to Genty,* always has at least 9 apparent 

 double points. The possible cases are then : — 



a) 6,9; a') 5,10; a") 4,11; a'") 3,12; a ,v )2,13; a v )l,14; a VI ) 0, 15 



b) 5,9; b') 4,10; b") 8,11; b"') 2,12; b IV ) 1,13; b v ) 0,14- 



c) 4,9; c') 3,10; c") 2,11; c'") 1,12; c IT ) 0,13; 



d) 3,9; d') 2,10; d") 1,11; d'") 0,12; d IV ); 



e) 2,9; e') 1,10; e") 0,11; 



f) 1,9; f) 0,10; 

 g)0,9. 



We shall describe in detail the septimic curves that have multiple points. 



1. A septimic curve can have a quintuple point provided no three of 



the tangents lie in one plane. Such a quintuple point is equivalent to 



* Bull, de la Soc. Math., t. 9, 10 (1881), p. 153. 



