into an edge of the circumscribed cone of SI hav*ing as vertex; 

 i> becomes tiie point of contact of tliat edge with £2, thus a point 

 of the intersection of i2 with the first polar surface of 0. Let us 

 imagine a point A^ of ?\, lying in the immediate vicinity of a branch 

 point, then tVom this point among others two generatrices of ii lying 

 very close together will start; the planes through those genera- 

 trices and are two tangential planes of the circumscribed cone 

 lying xery close together, and OAi is therefore a line lying in the 

 immediate vicinity of that cone. At the transition to the limit the 

 branch point becomes, just like the point /i mentioned above, a point of 

 intersection of 12 with the first polar surface of 0. This inter- 

 section, however, in our case breaks up into a number of separate 

 parts. Through a double edge of S2 e. g. pass two sheets of i2 and 

 passes 07ie sheet of the first polar surface; the double edge forms 

 thus a i)art of the intersection of the two surfaces, counts however 

 double, and it furnishes therefore in its point of intersection with 

 i\ two coinciding branch points. Of course likewise for }\. 



Suchlike considerations iiold also for the n {n^ — 4) cuspidal edges 

 of i2. Each plane through cuts S2 according to a curve having 

 cusps on the cuspidal edges, and it is well known that the first 

 polar curve of O with respect to that curve contains the cusps and 

 touches the cusi)idal tangents. From this ensues that the first polar 

 surface of 0, with respect lo ii, contains the cuspidal edges, and has 

 in each point of such an edge the tangential plane in conmion 

 with £2 ; each cuspidal edge counts thus three times for the 

 intersection and furnishes also three coinciding branch points on 

 ]\ and i\. 



All bi-anch points have been accounted for in this way. 



§ 8. The apparent circuit of the surface i2 out of an arbitrary 

 [)Oint of space on a plane e.g. is the section of that plane with 

 the projection (out of as centre) of the intersection of S2 with the 

 first polar surface of 0. This intersection consists however, as we 

 already saw in §7, of a number of separate parts. For i2 the directors 

 7\ and i\ are n (7i— l)-fold lines, for the polar surface | /i (?i — 1)— 1|- 

 fold lines; for the intersection of both they count 7i(/i — l)\?i{n — 1) — 1| 

 times. Each of the {n-\-l){7i)(n — 2){n—'S) double edges counts twice, 

 each of {he%i {ii- — 4) cnspidal edges three times, and as the complete 

 intersection is of order ^n {n — l)\2n{n — 1) — Ij, there remains a 

 proper curve of intersection of order 



2^i(n-l)|2^n-l)-l|-2/i(nl){n(/»-l)-l|-2(n f l)(w)(n-2)>-3) -3«(«'-4) = 

 2n' — 9?i' + 10;r -f lOn — 12. This is thus at the same time the 



