( 551 ) 



the pencil ABEF to pass through C and I), [f we take ABCDEF 

 for the conic of the pencil ABC/), then A" is indefinite on i£F, 

 whilst point L is to be found somewhere in L. L on f///. The corre- 

 spondence between the points K and A is of such a kind that to a 

 point L differing from L. 2 the same point K always corresponds, 

 namely K lt whilst when L coincides with A 3 point K is arbitrary 

 on EF . So the conic N breaks up into the two points K 1 and />.,. 

 The relation between the conies of the pencil ABCD and the tangents 

 KL or PP' of N is of such a kind, that to the conic ABCDEF 

 every line through A 2 corresponds and that, for the rest, between 

 the conies ABCD and the lines through K t a projective relation 

 exists, in which to the conies ABCDEF, ABCDG, ABCDH and 

 the degenerated conic AB . CD respectively K x L i} K X G, K^H and 

 K X L X correspond. From this is also evident, that the curve ( ', 

 breaks up into the conic ABCDEF and a C 3 passing through 

 A, B, C, D, G, H, K x and L x and farther that C 3 passes through the 

 points of intersection of K X L 2 with the conic ABCDEF. 



The double points of C, = AB . CD . ABCDEF. (', differing from 

 the base-points are K l} L x , T and the two points of intersection of 

 K X L. 2 with ABCDEF. The latter two doublepoints do not furnish a 

 triplet of points through which conies of the three pencils pass, but 

 two coinciding pairs of points; the branches through one doublepoint 

 correspond to the branches through the other and, it goes without 

 saying, in such a way that the branches belonging to C 3 corre- 

 spond mutually and likewise the branches belonging to the conic 

 ABCDEF. 



11. If moreover the points A, B, C, D, G and H lie on a conic, 

 C\ breaks up into that conic and the line K X L X {L. 2 then coincides 

 with L x ) so that the locus proper then consists of the conies ABCDEF 

 and ABCDGH and the lines AB, CD and h\ L x . When conic ABCD 

 does not pass through E, and F neither through GandH, the point 

 K coincides with K x and L with L x ; so that the pair of points PP' 

 lying on that conic is always determined by the same line K X L X . 

 Hence K X L X forms part of the locus. The C 7 has now seven double 

 points differing from the base-points, namely one triplet A",, L x , T, 

 and two pairs, the two points of intersection of K x L x with the conic 

 ABCDEF and those with the conic ABCDGH. 



If the point K x coincides with L x and therefore also with T, i.o.w . 

 if the four lines AB, CD, EF and GH pass through one point, on 

 each conic of the pencil ABCD the two involutions coincide. The 

 locus proper then becomes indefinite. If ice briny through an arbi- 



