102 



An elementary curve is said to be of the n^^ order, when lines 

 exist which have n, but no lines which have more than 7i points 

 iji common with the curve (unless the curve includes the entire line). 



In this note we chiefly consider elementary curves of the third 

 order. Some of the results obtained by Juel which shall prove most 

 useful are the following: 



The possible forms of elementary curves of the third order are: 



1. One connected curve of the third order wit hout double point orcusp. 



2. One connected curve of the third order with a cusp (the two 

 branches arrive at the cusp from different sides of the tangent, cusps 

 where the two branches meet from the same side cannot exist on 

 curves of the third order, as a slight change in the position of the 

 tangent would produce 4 points of intersection). 



3. One connected curve of the third order with double point, 

 (this variety can be considered as composed of a curve of the third 

 order and one of the s'econd ^) having only the double point in 

 common and each forming an angle at that point). 



4. One connected curve of the third order and one of the second ^) 

 (that is : oval, boundary of convex region) having no points in common. 



5. One connected curve of the third order and isolated point. 



6. Straight line and oval ^). 



7. Straight line and isolated point. 



8. Three straight lines. 



As points of intersection with a line are counted : 



double: ordinary point (that is: internal point of a convex arch) 

 on the tangent, isolated point on every line, cusp on every line 

 except the tangent and double point on every line except on either 

 of the tangents. 



triple: point of inflexion on tangent, cusp on tangent and double 

 point on both tangents. 



All other modes of intersection are counted single. 



We define as elementary surface of the third order F' any set 

 of points in the projective R^ possessing the two following properties ^) : 



1) These curves of the second order of course need not have finite breadth, but 

 can have one or two points in common with the line at infinity. (We always 

 consider projective space). 



2) Ultimately it may be advisable to make tliis definition less restricting. In order 

 to admit conical points it will be necessary to extend the first condition and to 

 make it possible that the surface degenerates both conditions have to be revised. 



The ultimate definition must be couched in such terms that no essential altera- 

 tions are required for defining elementary surfaces of order higher than the third. 



