524 PROCEEDINGS OF THE AMERICAN ACADEMY. 



V (10) as obtained from a IV (8), we add only five actual double points, 

 for the branches of the sextuple point whose tangents lie in one plane 

 are considered as having one connection in the limit with the branches 

 whose tangents lie in the other plane. 



Classification of Quintic, Sextic, and Septimic Curves 

 that have multiple points. 



A. 



Quintic Curves. 

 A quintic curve, according to the enumeration of Salmon,* can have 



a) 2 and 4 f ; a') 1 and 5 ; a") and 6 ; 



b) 1 and 4 ; b') and 5 ; 



c) and 4. 



In addition to these there is the quintic that has a triple point at which 

 the tangents do not lie in one plane. | This quintic can have no actual 

 double point in addition to the triple point. It can be obtained as the 

 partial intersection of the quadric cone ab — d 2 and the cubic monoid 

 abd + b 2 c + (b 2 + d 2 ) e that have the point ab d as the common vertex. 

 The residual intersection is the line b d. The point abd being a quad- 

 ruple point on the complete intersection and an ordinary point on the 

 line b d is a triple point on the quintic curve. As the tangent lines at 

 the triple point are the three lines that in addition to the line bd form 

 the complete intersection of the quadric cones a b — d 2 and b 2 + d 2 , it is 

 evident that they do not lie in one plane. § As the quadric cone ab — d 2 

 can have no double edge, the quintic curve can have no apparent 

 double point when viewed from the triple point. It has, therefore, 

 mk — fc (fc -j- 1) = 5 • 3 — 3*4 or 3 apparent double points when viewed 

 from an arbitrary point. The triple point on the quintic is equivalent, 

 as we have seen, to two actual and one apparent double points. The 

 quintic can therefore be obtained directly from a) above, as there are 



* Salmon's Geometry of Three Dimensions (1892), p. 318. 



t The two numbers in each case correspond to the number of actual double 

 points and apparent double points, respectively. 



\ Salmon's Geometry of Three Dimensions (1882), p. 320. 



§ No two of the three lines can coincide,' as the plane through this line and the 

 third line would meet the curve in six points. 



