744 



('lianu'lei- of tangent plane, only we had no certainty with regard 

 to the curves in planes thi-ongh a. Now all possibilities have been 

 considered it appears that for no point A two different planes can 

 pass through a both possessing one of the examined characters (we 

 obtain an immediate contradiction bj considering a plane through 

 .1 not containing a). It follows that in the three above cases no 

 plane through a {=\— (t) can contain branches departing from A (except 

 a itself). This completes the demonstration that « is tangent plane. 



Theorem 2 : If A moves continuously along a, then the tangent 

 plane also changes in continuous fashion. 



Let the points A^, A^ . . . . on a converge towards A. Tangent 

 planes a^, a^ .... a all passing through a. We assume that «,, «,.... 

 have a limiting plane a' different from a and shall prove that this 

 leads to a contradiction. Let ^j,, ■?,.... ,i be planes respectively 

 passing through A^, A, .... A and all ± a. The line of intersection 

 of «1, and |J, we denote by b^, the one of «, and /?, by 6, etc. 

 Lastly let h be the lijie of intersection of « and /? and h' the one 

 of ({' and ^. According to our assumptions h' and h do not coincide 

 and // is the limiting line of h^,b^ . . . . 



Now b is tangent at .4 to the curve in (^ and in the converging 

 planes i^i, ii.^ .... the curves have for tangents at A^, A^ . . . . 

 respectively the lines b^,b^ . . . . converging towards b' in ^. 



According to theorem 2 of the second 

 communication the curve in jiis the limiting 

 set of the curves in (i^, ^^ .... (with the 

 possible exception of an isolated point). 

 Let c and d be lines through A m ^ 

 separating b from b' . The corresponding- 

 planes through a shall be denoted by y and rf. 

 For n large enough a bi-anch departs 

 in /?„ from A,, in both directions inside 

 those opposite angles between y and din 

 which b' is situated. Loops contracting 

 towards A are evidently excluded, hence in order that in /i no branch 

 departs from A inside those angles of c and d which contain //, it 

 is unavoidable that in the converging planes the above mentioned 

 branches leave (hese angles via points of the planes y and 6 (or 

 one of these) converging towards A. Hence in at least one of the 

 planes y and 6 the point A would be limiting point of points of 

 F^ not situated on a. This means tlrat in one of those planes a 

 branch departs frouj A different from a, but this is a contradiction 



Fis-. 14. 



