309 



tangent will contract towards A. This means (hat in the plane J the 

 point A counts at least double on (/, but <I has two other points 

 in common with 7^' : a contradiction. 



3. A is cusp V with b {=\= a) for tangent. The original semiplanes 

 « and /?, in which A is cusp are again supposed to lie below y. 

 Semiplanes through a converging from below towards the semiplane 

 of 7, in which A is isolated, again show ovals through .4 with a 

 for tangent and contracting towards .1. Let c be a line in y through 

 A (=1= rt and =\= b). The contracting ovals show that c is tangent 

 at A in every plane (=!= y). Besides in all these planes A is ordinary 

 point because c has still another point in common with F'. c is 

 an arbitrary line in y through A (—1= a or ='= b), hence in every 

 plane through A (except those through a or b) the curve is, in the 

 vicinity of A, situated below y and the tangents at ^ all lie in y. Let 

 us now consider one of the original semiplanes with cusp in A, for 

 instance a. Applying the same reasoning given on p. 115 — 116 it is 

 found that every line through A in a (except a and the cuspidal 

 tangent in a) must be tangent at A in every plane passing through 

 that line (except «). But this contradicts the result obtained above 

 that in every plane through A (except those through a and b), A 

 is ordinary point with tangent in y. 



^ 6. Through A passes at least one plane in which A is either 

 isolated, double point or cusp. 



Let A be ordinary point in two different planes, such that the 

 tangents a and b in A do not coincide. In the preceding pages it has 

 been shown that when A counts double on a line in a plane, then 

 A also counts double on that line in any other plane. From this 

 follows that in the plane through a and b, A counts double on both 

 these lines, hence in that plane A is either isolated, double point or 

 cusp. Remains to prove that through an arbitrary point A of F' 

 pass two planes in which A is ordinary point with non-coinciding 

 tangents. Except isolated points, double points, cusps and ordinary 

 points there are only points of inflexion. We begin by showing that 

 not all planes through A can have a point of inflexion at A. 



Let a be a line through A which has still another point in 

 common with F\ and thus can never be tangent at a point of 

 inflexion. Suppose every plane through A shows a point of inflexion 

 at A. In every semiplane through a departs from A a convex arch, 

 situated at tlrst either above or below the tangent at A (at the 

 outset it was assumed that no line through A belongs entirely ot F*). 



