( 490 ) 



of intersection ^4^ with ?\. In liie plane ^i?'.^ iiow pass also through 

 A^, besides tlie double tangent, only 7i {7i — J) — 2 tangents to ^'S so 

 that now again on the line A^P^P, must lie 2, (n — 2) points of 

 intersection of h with the satellite curve of ;:»i"~'. In the n — 4 

 points of intersection of the double tangent with t' the satellite curve 

 of pi^'~"' will again touch t'; the missing four points must be divided 

 regularly among the two points of contact P^ and 1\, from which 

 ensues that the satellite curve of p^""^ touches the double generatrix 

 of S2 in P^ and P^. The satellite curve of c"" will thus also have 

 this pro|)erty ; however as regards c"" itself, it passes also through 

 7^1 and P^, but without touching the line A^P^P^ in these points; 

 so on all the double generatrices of £1 together lie 2 (7i-{-l)(7i)[n — 2){n — 3) 

 points of intersection of c"' with its satellite curve. 



Now 6'"- and its satellite curve have more points in common still, 

 but these lie all on )\ and ?\. The surface /7j has r, as a single line 

 (§ 3), on the other hand iJj has r, as an \?i{n — 1) — 2 |-fold line, so 

 the intersection of the two breaks up into a curve and the line /-j, the 

 latter counted \ii{n — 1) — 2| times. The surface cuts r.^ in the 

 n points >Sj ; so these count for n \n {n — 1) — 2 | points of inter- 

 section of the three surfaces <f>, U^, U^, and therefore for as many 

 points of intersection of C"'' with its satellite curve. We saw further 

 in § 5 that the satellite curve of c"', thus the intersection of '/' and .2'j, 

 has in the n points S^ on r^ again \n{n — 1) — 2 j-fold points; as *l* 

 contains these points also, they count for n \ n {n — J) — 2 | points of 

 intersection of c' with its satellite curve. 



We now add the different amounts found, thus 2/i (n' — 4), 

 2 (?? + !) {7t){7i—2){n^3), 2n\n:n—l)—2\ together, and we find 

 2/i' («^2j, just the complete number of points of intersection of the 

 three surfaces 0, /Zp 2^ of ordei' )i,?2,2n{n—2). 



§ 7. Through a point A^ of 7\ pass 7i{?i^l) tangents to the 

 curve k" lying in the plane A^7\ and these intersect ?\ in ti {n — 1) 

 points A^; inversely to such a point A^, 7i{n — 1) poinis A^ cor- 

 respond, from which ensues tliat we can regard the surface i2 

 as generated by the lines connecting the corresponding points of two 

 series of points lying on ?\ and i\, between which there is a 

 \n{n — 1), n(n — l)|-correspondence. If we project these two series out 

 of an arbitrary line /, then two coUocal pencils of planes are formed, 

 between which there is likewise an \ti {ii — 1), 7i (>i— l)|-corres{)ondence ; 

 the 2/z (>i — 1) coincidences are planes each containing the line con- 

 necting two corresponding points, thus a generaliix of i2, out of 

 which follows 2n {n — 1) for the order of 52 (§ 2). 



