107 



On X'^\ A' and A" forn a pair of' an involution; of' the straigiit 

 lines i':=:A'' A" six pass thron<;li .1. Three of them are indicated 

 by the intersections A' of / and a^ ; here A' lies every time in A. 

 The remaining three ai'e the lines 6^. ; for each of them contains a 

 pair A'', A'" corresponding to the point X^^{/h„i„)- 



The curve (,<;)o envelojied by .i' is rational, because we can associate 

 .!■ to A'; it has therefore ten biiaiujents. As suci) a bitangent bears 

 two pairs A', A" and V' , Y" it follows that the involution {X,Y) 

 contains ten pairs on /, and conse(|uently is of the tentli class. 



5. Let a straight line / be revolved round a point E; the pairs 

 A", X" and Y', Y" Ijing on it describe then a curve 8\ which 

 passes twice through E and is touched there by the straight lines 

 EE' and EE". On EA lie two points A'' and Y, each forming with 

 E a pair of the (A''"*) ; so .4 is a node of e\ For the same reason 

 t" has nodes in Bj^; it also contains the points C),. In consequence 

 of the existence of 5 nodes, s" is of class 20, so that i^J lies on 16 of 

 its tangents. Of these 8 contain each a coincidence of the (A^^); the 

 remaining 8 are represented by four bitangents, being straight lines 

 s, on which both pairs belonging to (A'") have coincided. From tiiis 

 it ensues that the lines .s- envelop a curve {s)^ of the fourth class. 

 Apparently the straight lines ,s', passing through A, are tangents to 

 «■'. In the same way the four tangents out of Bh to ^iJ" are the 

 straight lines s, which may be drawn through Bk- 



Apart from the singular points b^ and d' have 16 points in 

 common ; to them belong the 8 coincidences of which the supports 

 (I pass through E. The remaining 8 must be jwints A', coinciding 

 with the corresponding point X witliout J's |)assing through E; i.e. 

 they belong to the locus f.-,. of the points A, which complete the pairs 

 lying on f" into groups of (A^). 



As E lies on three of the straight lines x = X'X" belonging to 

 B/c, Bjc is a triple point of f,. ; ajialogously .1 and 6a are simple 

 points of that curve, so that the latter has 2 -f- 3 X 2 X «^ H~ ^ = '^^ 

 intersections witli 5" in the singular points. Besides the 8 points of 

 rf^ indicated above they have moreover the points E', E" in common; 

 so we conclude that f,|. must be a curve of the sLvt/i order. To the 

 intöt'sections A of t\^ and / coi'res[)ond lines .t;, which iias» through 

 E; from this it ensues again that ./; envelops a curve of the sixth 

 class, when A' describes the straight line /. 



6. If E is laid m C\, t" is replaced by the tigure composed of 

 the singular conic Yi' f^"^' ''^ curve 7/, which has a node in C\, 

 and passes through the points A,Bk,Ck. The two curves have apart 



