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through A and B contracts to the point where A and R meet. The 

 most rational thing to do is to consider this meeting point as a 

 special kind of oval in the tangent plane. We can also imagine 

 that the oval through A and B contracts to a segment of a. All 

 points of this segment would have the same tangent |)lane (tangent 

 plane of the first kind, examined at the beginning). Now the admis- 

 sion of this possibility has the disadvantage that we should be more 

 or less forced to consider the linesegment in the tangent plane as 

 a special sort of oval and going back to the definition of elementary 

 curves we should not ordy have to admit isolated points, but line- 

 segments also. This would cause the development of the theorj' to 

 become a good deal more complicated but the enlargement of the 

 results would probably remain very trivial. To mention an example ; 

 to the elementary surfaces of the second order would be added the 

 plane convex regions including the boundary and the linesegment. 

 Far greater would the changes become if we also dropped the 

 condition that the convex arc is not to contain linesegments. This 

 however would mean an entirely different problem. 



