( 491 ) 



On each of the two bearers lie 2;i(^—l)|n(?i-l)— 1| =: 2n(?i'-2?i'-|-l) 

 branch points M, i. e. points of whose corresponding points on the 

 other bearer two coincide, which coinciding points are then called 

 double i)oints; we shall now investigate how in our case the branch 

 points put in an appearance.. We consider therefore in the first place 

 the 71 points of intersection S^ of r^ with <P. In the plane S^i\ lies 

 a curve X" passing through S^ ; so through S^ pass n {n--l) — 2 

 tangents which do not touch in S^, and two coinciding ones which do 

 touch in S^ ; so evidently S^ is a branch point on i\, and the point 

 of intersection of the torsal line passing through S^ with 1\ is the 

 corresponding double point. Number n. 



Through i\ pass 7i{n—\y tangential planes of «2>, and each of these 

 cuts fp in a curve X" with a node. If the point of intersection 

 of such a plane with i\ is a point A^, then out of A^^ start 

 n{ii — 1) — 2 proper tangents to k', whilst the line connecting x4i and 

 the node counts for two coinciding ones; so A^ is also a branch 

 point. Number ii{n — J;\ 



Further in § 4 we found n{ii'' — 4) generatrices of i2 which are 

 at the same time princijial tangents of *P. If the point of contact of 

 such a principal line with is P and A^ the point of intersection 

 of the plane Pi\ with i\ , then from A-^ start n {n — 1) — 2 ordinary 

 tangents to Z," and moreover the inflectional tangent A^P to be 

 counted twice; so A^ is again a branch point. Number n{n^- — 4). 



Finally in § 6 we fo\xi\& {ii-{-l) {ii) {ii—2) [n — 3) double generatrices 

 of i2; it is clear, that also the points of intersection of these with 

 1\ and i\ are branch points. Number {ii-\-'\) {n) [n — 2) (??— 3). 



Other branch points there are none. If e. g. a point A^ is to be 

 a branch point, then two of the tangents out of A^ to /;" must 

 coincide, and that is only possible in one of the four ways described 

 above. If now the four mentioned numbers are added up we do 

 not find the required complete number of branch points 2n{n^ — 2n'-\-l), 

 but only n{if — 2if — /i -j- 4), i.e. for very great values of n only 

 half; on the other hand we find the exact number, if we bring the 

 n{n'^ — 4) points of the third group three times into account, and the 

 {n -f- 1) {n) {n — 2) {n — 3) of the last twice. The question is how 

 to explain this. 



If we bring a plane through an arbitrary point of space and 

 a generatrix b of i2, and likewise through an adjacent generatrix 

 6*, and if we then let b tend to b* to coincide with it finally, then 

 at the limit the line of intersection OBB* of the two planes passes 



1) Emil Weyr 'Beitrage zur Curvenlehre", S. 3. 



