( 549 ) 



wish to treat the ease more closely, thai one of the pencils lias two 

 points in common with each <>t' the two others, where we shall 

 attain at results in another way, which will prove !<> agree to the 

 general ones and complete these in some parts. 



Let ABCD, ABEFa,nó CDGH be the three pencils of conies. On 

 one conic of the pencil ABCD the two other pencils describe two 

 quadratic involutions of which the connecting lines of the pairs of 

 points pass through a point K of FF, resp. a point L oi' GH. The 

 pair of common points PP' of these two involutions is thus deter- 

 mined by the right line KL. If the conic AB CD describes the whole 

 pencil, K and L describe projective series of points on EF&nó (ill. 

 For, if we take K arbitrarily on EF, the conic ABCD is determined 

 by it, as it must pass through the second point of intersection of 

 CK with the conic ABEFC (as likewise through the second point 

 of intersection of DK and the conic ABEFD); by the conic ABCD 

 the point L is unequivocally determined. Reversely to a point L 

 of GH now corresponds one point K. The projective series of 

 points are however in general not perspective; so the line KL or 

 PP' envelops a conic K touching EF and GH. 



Of that conic three other tangents are easy to construct, namely 

 by taking for the conic ABCD in succession each of the three 

 degenerations. If that conic is AB . CD then the movable points of 

 intersection with conies of the pencil ABEF lie on CD so that K 

 lies on CD, thus in the point of intersection K x of CD and EF , 

 likewise does L coincide with the point of intersection L l of AB 

 and GH. The line K X L X is thus tangent to JSf. The construction 

 becomes a little less simple if we take one of the other degenerations 

 e.g. AC.BD. By cutting this by the degenerated conic AE . BF 

 of pencil ABEF it is evident that K coincides with the point of 

 intersection of EF with the line connecting the point of intersection 

 of AE and BD with the point of intersection of BF and AC; in 

 similar manner L is found. 



To the locus of the points P and P' belongs the locus of the 

 points of intersection of the conies of the pencil ABCD with the 

 prqjectively related series of tangents KL of the conic N~. This locus 

 (as is easily evident out of the points of intersection with an arbitrary 

 right line or with an arbitrary conic of the pencil ABCD) is of 

 order live with double points in A, B, C and D ; further it passes 

 through E, F, (t and H, as K coincides with E when the conic 

 ABCD passes through E, etc. If we take for the conic of the pencil 

 ABCD the degeneration AB . CD, then KL passes into K x L l which 

 line cuts the conic AB . CD in the points K x and L lf which thus 



