983 



of Reyk, is (oiiiid as follows. We consider two pencils of conies, 

 which have a coniinon base-point E ; the reniainini-- base-points we 

 call F„F,,F, and (;„(i,,G,. If each conic throngh 7^, i^'i is brought 

 into intersection with each conic through E,(h, a {P^) is acquired, 

 possessing a singular point of the fourth order in E, and singular 

 points of the second order in FkJh ') 



If, however, the points Gk lie on the rays EFi, then the dege- 

 nerate conies {EF„FJ\) and {EG,, G,G,) have in common the line 

 h, = F,G, and the point H, = {F,F,, G,G,) ; now H, is a singular 

 point corresponding to all the points of k, ; consequently it is in the 

 same condition as the point C mentioned above. There are now two 

 more similar points still, H, = {F,F„G,G,) and H, = iF,F„G,G,). 

 While with an arbitrary situation of the points F and G, a q' 

 corresponds to a straight line r, which q' passes four times through 

 E and twice through Fk, Gk, this curve degenerates now into the 

 three lines hj, = FkGk and a q\ which has a node in the third 

 vertex D cf the triangle of involution, of which ?• is a side. On this 

 o\ P' and P" are now again collinear with D, so that ?i = l. 

 ' If G, is placed on EF, and G, on EF,, a special case ofa(P=^) 

 is found, where n — 2. The curve q' now loses only the straight 

 parts /ii and A, , consequently becomes a q' having nodes in E, F,, 

 G, and D; on this quadrinodal </, {F , F') form again the involu- 

 tion of pairs, so that n appears to be 2. The singular points of the 

 second order are E, F^, G„ the singular points of the first order are 

 F ,F\,G„G^,H„H^; but the last two have respectively been asso- 

 ciated 'to all the points of h, and h„ while to each of the first four 

 a quadratic involution cori-esponds. 



13. In the case n = 3 we have the relations 



2 {m—iy = 8 and « + ^m'- = 28. 



The first holds in three ways, for 



8 =: 2 X 2-^ = 2^ + 4 X 1' = 8 X 1'. 



But the first solution must be put aside at once. For by (P,P') a li"e 

 r would be transformed into a q' ; for the connector of two singular 

 points of the 3'^ order q' would have the two corresponding curves 

 {By as component parts ; but then there would be no figure corre- 

 sponding to the remaining points of the line in question. 



The third solution too must be rejected, as, for 8 singular points 

 of the second order f« + 8 X 2'^ = 28; so « i= - 4 would be found. 



1) See my paper, relerred lo above, in volume XII I of these Proceedings (p.p. 90 

 and 91). The notation has been altered here. 



