( 552 ) 



trary 'point I' a conic oj each of tin pencils, then those conies have 

 another second common point, namely the second point of intersection 

 of the line TP "'it// the runic ABCDP. The envelope proper is then 

 still definite and consists of two coinciding points T. 



12. If tin' points E and G coincide, then if the conic of the 

 pencil ABCD passes through E the point K as well as the point A 

 coincides with E. The series of points A' and A are perspective, the 

 lines KL all pass through a selfsame point /'. 



The conic N breaks up into two points A' and U. As E belongs 

 to the part improper of the envelope, the envelope proper now consists 

 only of point U. By taking- lor the conic of the pencil ABCD the 

 degeneration All. (I) it is evident that I' lies on the line K X L X . 

 Another line KL and by that the point U itself can be constructed 

 in the way indicated in § It by allowing the conic ABCD to break 

 np into AC.BJ) or A I) . IK'. 



Between the lines KL or PP' through U and the conies of the 

 pencil ABCD exists a projective correspondence, where to the conies 

 ABCDE, ABCDF, ABCDH and AB.CD respectively the lines 

 UE, UF, I'll and K l L 1 correspond. TJie locus of the points of 

 'tula-section is <i cubic through the points A, I!. C, l> U, E, /•'. //. I\\ 

 and L x , which is determined by these 1<> points ; the third points 

 of intersection of that curve with AC, AD, BC and III) are easy 

 to construct. 



On the conic ABCDE the two involutions coincide, so that that 

 conic has separated from the C s of § and has become improper. 



The locus proper consists note of tin' lines All and CD <tn<l the 

 above-named C 9 , so it is of order five. Differing from the l>ase- 

 points the C s has three double points, A,, L x and 7' (the point of 

 intersection of AB and CD) forming a triplet. 



If moreover the points .1, A', C, I), A' and F lie on u conic, no 

 other particularity appears than the point I' coinciding with l\\. <>f 

 the three points of intersection A',, A : and U of k\ L x with the ( \, 

 the points K x and U now coincide, so that the (\ touches tin' line 

 ÜT, L x in K x . In comparison with § 10 the particularity that appears 

 is this that the point L., coincides with K whilst the pencil of rays 

 L 2 has passed into the part improper of the envelope and the conic 

 ABCDEF into the part improper of the locus. 



13. The case treated in the preceding paragraph is of course not 

 the only one in which the series of points A" and L are perspective, 

 the condition of that perspectivity being single, the condition of the 



