532 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Here fi = 4, v = 2, k = 4, k f — 1, A' = 0, H" = 3 ; therefore fl = 9. 

 This septimic may have as many as three actual double points in addition 

 to the triple point. 



A septimic curve with a triple point at which the tangents lie in one 

 plane can be obtained also as the partial intersection of a quartic cone 

 and a quartic monoid. We can then obtain the following species : — 



(1). Those having 9 apparent double points ; these may have as many 

 as three actual double points in addition. 



{2). Those having 10 apparent double points; these may have as 

 many as two actual double points in addition. 



(3). Those having 11 apparent double points; these may have one 

 actual double point in addition. 



(4)- Those having 12 apparent double points. 



These cases can be obtained directly from d), c), b), a), c'), b'), a'), 

 b"), a"), a'"), respectively. 



8. A septimic curve having a triple point at which the tangents do not 

 lie in one plane can be obtained as the partial intersection of a quartic 

 cone and a quartic monoid. We shall then obtain species similar to the 

 four above. These different septimics can be derived directly from 

 d')» c ')> b ')> a ')> c ")> b ")> a "); b '")> a '")> a ' v )> respectively. 





