( 554 ) 



nine former points 1 ) and thus the same as C 3 through the ten poinis 

 AJl,(\l),E,E,(i,K x ,E x and //. The line VG would then however 

 have four points G, L ly H and H' in common with this G 3 . 



So we arrive at the following simple result: 



If the ten points A, B, C, I), E, F, G, H, k\ and L l lie on the 

 same cubic, the series of points K and L are perspective, whilst the 

 cadre of perspectivity coincides with the third point of intersection U 

 of K l L l with C t . The envelope proper brcals up into the point U 

 and the point of intersection V of EF and (ill. The locus proper 

 consists of the lines All and ('I), the cubic just mentioned and the 

 conic through A, B, C, 1) and the two points, in which the right line 

 UV intersects moreover the C t besides in U. 



If E and G coincide, we immediately see that the above condition 

 is satisfied. The point V lies then in point E so that one point of 

 intersection of UV with C t differing from V becomes the point E; 

 the indicated conic is thus the conic ABODE, which now however 

 belongs to the part improper of the locus. 



14. If G coincides with E and II with F, then the series of 

 points K and /. are connective with double points in E and F. 

 The pair of points PP' on an arbitrary conic of the pencil ABCD 

 is now continually described by the same line EF, thus belonging 

 to the locus proper. If the conic passes through E or F the two 

 involutions coincide, so that the conies AB CDE and AB CDF belong 

 to the locus, but to the part improper of it. Moreover the lines All 

 and CD belong to the locus proper, so that the tatter consists of the 

 three lines AB, CD and EF. An envelope proper is no more at 

 hand, the line connecting P and P' coinciding with AB, CD or EF 

 when T and P' differ from the base-points. 



In comparison with § 12 the particularity appears that /'coincides 

 with F, that the pencil of rays U passes into the part improper of the 

 envelope and that the C 3 breaks up into the conic ABCDF becoming 

 improper and the right line EF. 



The case of the pencils of conies ABCD, ABEF and CDEFcm 

 be profitably used to define with the help of the principle of the 

 permanency of the number the order of the locus of P and P' and 

 the class of the envelope of PP' for the case of pencils of conies 

 lying arbitrarily with respect to each other. Starting from this simplest 



l ) The C 3 is only then not determined by these nine points if two of those 

 points coincide in such a way that the connecting line is indefinite (e. g. G with 

 E or K x with L{). Then the ten points lie on a C 3 , whilst it is easy to prove that 

 the correspondence between K and L is perspective. 



