Surface reciprocal to a given one. 483 



face at a point in a double line, each of these points counts for three among the 

 intersection of the three surfaces. The points then not on the double line in 

 which the three surfaces intersect is n|(?z— 1)^— 1 j — 3 x (2m — 4). Or the 

 double line diminishes the degree of the reciprocal by 7?i — 12, as we proved 

 otherwise, p. 467. Or again, the intersection of the given surface with one of 

 the polar surfaces consists of the double line, and of a curve of the degree 

 n(n — 1) — 2, meeting the right line in 3?i — 6 points. For a surface Ax^ + Bxy 

 + Cy' having a double line, meets any other Dx + Ey passing through that line, 

 in a curve which meets that line in the ot + 26 — 4 points where the line meets 

 AE^-BDE+CD\ Of the 3?i-6 points, 2w-4 are the cuspidal points, the 

 remaining n— 2 are the points where the line meets a certain plane section. 

 And the points of intersection {not on the double line) of the three surfaces 

 are (?i — 1) jji(n — 1) — 2| — 2(2w — 4) — (ft — 2), as has been already found. 



In general, if a surface have a right line of the degree p of multiplicity, 

 this will be a, (p—1) multiple line on each of the two polar surfaces, which 

 will intersect besides in a curve of the degree (n — 1)'- — (p — l)-. And the 

 latter curve will meet the right line in the 2 (p— 1) (n—p) points of special 

 contact previously noticed. It will, therefore, meet the surface in points not 

 on the multiple line n\(n-iy — (p—iyi— (p—l)2(p—l) (n—p). The mul- 

 tiple line, therefore, diminishes the degree of the reciprocal by (3^ + 1) (p — l)n 

 — 2p(p^-l). Or again, the original surface intersects any polar surface in a 

 curve of the degree n(n — l)—p(p—l) meeting the right line in the 2 (p—1) 

 (n—jo) points of special contact, and in n-p other points. And the three 

 surfaces will intersect in points not on the multiple line, 



(n-l)\?i(n-l)-p(p-l)\-p.2(p-l) {n-p)-(p-l) (n-p), 



which agrees with the result obtained already. 



I next investigate the diminution produced by the multiple line in the 

 number of cuspidal and double edges of the cone of simple contact. 



The cuspidal edges answer, as has been before proved, to the intersections 

 of the surface with a first and second polar surface ; and the multiple line is 

 of the degree jo — 2 on the latter surface. I have satisfied myself that the for- 

 mula for the number of intersections of the three surfaces is, 

 (n-2)\n(n-l)-p(p-l)\-(p-2) 2(p-\) (n-p)-(p-l) (n-p). 



