( 553 ) 



coincidence of E and G being double. The condition of perspective 

 can he found out of I lie condition that the point of intersection V 



of E F and Gil (as point K) corresponds to itself (as point /, . 

 Now the conic A BCD belonging to V (as point K) passes through 

 the second point of intersection W of CV with the conic ABEFC, 

 whilst C,W is a pair of points of the involution described on 

 the conic ABCDW by the pencil ABEF. If this pair of points 

 also belongs to the involution described on that same conic by the 

 pencil CDGH, the point L coincides evidently with V. So this 

 is the case when the conic of (he pencil CDGH touching the conic 

 ABCDW in C passes through W. This condition for the perspec- 

 tivity of the series of points K and L (where of course it must be 

 possible to interchange C with D and likewise AB resp. EF with 

 CD resp. GIF) is evidently satisfied when E and G coincide. 



If i is the centre of perspectivity, there exists between the rays 

 of the pencil U and the conies of the pencil ABCD a projective 

 correspondence, where to the conies ABCDE, ABCDF, ABCDG, 

 ABCDH, ABCDW and AB . CD correspond respectively the rays 

 UE } CF, UG, FH, FV and K x L Xi whilst moreover to the conic 

 ABCDW all the rays of the pencil V correspond. So the C 6 of §9 

 breaks up into the conic ABCDW, still belonging to the part proper 

 of the locus, and a i\ passing through the points A, B, C, D,U, E, 

 F, G,H,K l and L 1} cutting the conic in A, B, C and D and the 

 two points of intersection of UV with that conic. 



The locus proper is thus a ('. consisting of the lines AB and CD, 

 the conic ABCDW and the C s before mentioned. This C 7 has live 

 double points differing from the base-points, namely, the triplet K iy L lf T 

 and the pair formed by the points of intersection of UV with the 

 conic ABCDW. 



The 6' 3 is determined by the ten points, A, B, C, D, E, E, G, H, K t 

 and E x so these ten points will have to lie on a C\ if the above 

 condition for the perspectivity is satisfied, and reversely it is easy 

 to prove that when those ten points lie on a C 3 the series of 

 points are perspective. Suppose namely that the series of points were 

 not perspective. Then it would be possible by keeping the points 

 A,B,C,D t E,F anti G to construct on the line GH (thus h\ 

 keeping the points K lt L lt Pand IT) by means of the former condition 

 for perspectivity a point H' in such a maimer that the series of 

 points K ami L are perspective ; FL' is then the second point of 

 intersection of 17/ with the conic through C, D, G and IT, touching the 

 conic ABCDW in C. So now the ten points J, B, C, I). E, I'. (>, K lt 

 E x and H' will lie on a C 3 , however already determined by the 



