( 550 ) 



lie on the locus of the points of intersection too. By taking the two 

 other degenerations we find four more points of C 6 . Altogether there 

 are 10 single and 4 double points by which C\ is determined. 



If we take the degeneration AB . CD, the particularity occurs, 

 that the pair of points of the involution described by the pencil 

 ABEF can become indefinite on AB, if namely the conic ABEF 

 breaks up into AB . EF. By this the whole line AB (and of course 

 the line CD too) will belong to the locus proper of P and P' l ). 

 To the part proper of the envelope of the lines PP' the pairs of 

 points PP' lying on AB or CD contribute nothing but the lines 

 AB and CD (which belong also to the part improper of the envelope, 

 the points A, B, C and D, which does not give rise to a higher class. 

 So the locus proper of P and P' consists of the lines AB and 

 CD and the curve C\ and is thus in accordance to the general 

 results of order seven. The line AB(CD) intersects (', in the points 

 A and B (C and D) to be counted double and in L l {K x ). The 

 curve C. has three double points differing from the base-points (of which 

 E, F, G and H are single and A, B, C and D threefold points 

 of C 7 ) namely l\\, L, and the point of intersection T of AH and 

 CD. These form a triplet of double points belonging together of which 

 we spoke in §5. The conies of the three pencils passing through 

 one of those double points, also pass through the two others; these 

 conies are AB . CD, AH. EF and CD . GH. To the branches 7'A", 

 and TL X of (\ passing through T correspond respectively the 

 branches K X T and L X T passing through K x and L 1} whilst the 

 branches of C\ passing through h\ and />, correspond mutually. 

 Summing up we find : 



For the conies A BCD, A HEF and CDGH the locus proper of 

 the pairs of common points HP' consists of the Zincs AB and CD 

 and a curve of order five, having in A, B, C and D double points 

 and in E, F, G and H single points and further passing through 

 the point of intersection K x of CD and EF and the point of inter- 

 section L x of AB and GH. The envelope proper of the lines PP' 

 is a conic touching the lines EF, GH and K X L X . 



10. If the points A, B, C, D, E and F lie on a conic, the 

 latter then belongs to the locus, so that the C b breaks up into that 

 conic and a C 3 passing through A, B, C, D, G, H, K x and L x . To 

 each conic of the pencil ABCD now belongs the same point K, 

 namely K x , as is immediately evident when we make the conic of 



} ) More generally : if two base-points of one pencil lie with two base-points of 

 another pencil on a right line, that line belongs to the locus proper. 



