As Ë is coUinear with 5 pairs') .Y', .Y" of a', and with two 

 pairs of /^z.', f.^^ has a quintuple point in A and nodes in Bk. 



If E is brought in A, eJ coincides with a\ 



For B^ e^^ consists of the curve /?/ and a curve *?l^ wiiich 

 passes three times throngii A and once through the 8 points Bk. 



The intersections A" of %' with the straight line / determine 8 

 lines ,v^X'X" passing through E; we conclude from this that .v 

 envelops a curve of the eighth class (/)8, when .Y describes the straight 

 line /. In confirmation of this result we observe that with the 8 

 intersections A' of I and «** correspond the 8 sti-aight lines passing 

 through A{X") to the associated points X'. 



As (/)8 must be rational, consequently possesses 21 bitangents, 

 I contains 21 pairs A', I", for which the corresponding points A^',A^"; 

 Y' ,Y" are coUinear. 



6. An arbitrary straight line contains three pairs (A'', A"'), (P, Y"), 

 {Z' , Z") of A'"'; the corresponding points A^, }", Z apparently form 

 a group of a new triple involution^), which we shall indicate by 

 (A YZ^ ; it appears to be of class 21 . 



Apparently (XYZ) has singular points in A and B/,. Let .v be 

 the order of the curve a, which contains the pairs Y, Z, belonging 

 to A' 17:^.4; let further y be the order of the corresponding curve 

 |ï'/.- belonging to Bk- 



Let the straight line / be described by a point Z, the associated 

 pair AT will tben describe a curve A, the order of which we shall 

 indicate by z. If attention is paid to the points of intersection of / 

 with a and ;?/,, it will be seen that ). must have an .r-fold point in 

 A, a _?/-fold point in Bi. 



In order to determine the numbers .r, ?/, z, we may obtain three 

 eqnations. 



We consider in the first place the intersections of the curves ;. 

 and n, which are determined by the straight lines /and?/?. To them 

 belong the two points which form a triplet with Im, further r points 

 Z, for which A^ lies on / and Y on m ; the remaining intersections 

 lie in the singular points. So we have the relation 



^^ = 2 + c + .r'' + 9y^ . (6) 



Let the curve a^ be described by Z, then the figure of order Sz, 



1) The curves y} and s^ have 3X5 + 9X2X2 = 51 intersections in the 

 singular points; they have 3 more points in common on E A; tlie remaining 10 

 intersections form 5 points X',X" coUinear with E. From tliis appears anew that 

 the curve of involution x^ is of class 5. 



-) This property is cliaractcristic of the triple involutions of the Ihlrd class. 



