936 
towards A when y converges towards the plane 8 through PQ and 
CF, for the alternative demands the existence of points on PQ, 
which can be points of inflexion with the tangent in 8, and this is 
impossible as the line CF belongs to #*. Hence a finite vicinity of 
A exists, inside which the converging planes contain no points of 
1 not situated on PQ. This means that along AF the sectors / and 
/V meet from the same side of 8, in other words CF counts double 
in 8. From this it follows that CH is not intersected by any line, 
not passing through A. In the same way as above we find the 
total number of lines on F* to be 7 (CF counted double). A is 
biplanar point, such that the line of intersection of the tangent 
planes lies on the surface, and one of the tangent planes at A is 
tangent plane all along this line. This case is excluded in what 
follows and we leave undecided if it can occur. 
§ 5 On surfaces F* on which the singularities described above 
do not occur. 
Except points lying on one line or no line at all, the surface can 
contain what we called normal points of intersection of 2 or 3 lines, 
and of a single and a double line. 
In comm. 3 p. 736—744 we proved (theorem 1): that every point 
on a line of F* has a tangent plane, provided this line is not inter- 
sected by any other. Above (§ 2) we already did some supplementing 
and brought forward two kinds of uniplanar points. Apart from this, 
the demonstration can, with some small self evident alterations’) be 
used to establish the following theorem: Lf a line of F'* counts double 
in no plane, then every point of this line has a tangent plane provided 
no second line passes through that point. 
Theorem 2 of p. 744 (comm. 3) holds, with the same demon- 
stration, when normal points of intersection are admitted. 
We pass on to comm. 4 (p. 1246—1253). As the above described 
singularities are excluded, theorem 1 (p. 1246) holds and can even 
be extended to the following: The lines of F* passing through one 
point, lie in one plane. 
Apart from the occurrence of a plane « in which the curve is 
composed of a double and a single line, the rest of comm. 4 can 
1) Example of a necessary alteration: p. 739, J. 6 from bottom: If the rest- 
curve consisted of 2 lines, these lines would have a normal point of intersection 
at B. Hence in 2 the curve would have 0 for tangent at B, and this cannot be 
as A is cusp in 2. If the restcurve in « is a double line through B, then a slight 
turning of 6 round A in 2 would replace B by 2 points of intersection and this 
again is impossible, 
