(52) 



Mathematics. — "On the rank of the section of two algebraic 

 surfaces.'" By Dr. W. A. Yersliys. (Communicated by Prof. 



P. H. SCHOUTE. 



1. Ill this paper I intend to prove the relation new to me 



r r= wij n^ -\- m^ n^ — 26 — 3/, {A) 



where r is the rank of the curve of intersection s of two algebraic 

 surfaces >S'i and S^, respectively of the degree n.^ and n^ and of the 

 class ???i and m^ and possessing in d points an ordinary contact 

 and in -/ points a stationary contact. Some applications of this 

 formula are given too. 



Formerly I proved ^) the following extension of a well-known 

 formula ') 



r = n^n^{n,-\-n.,— 2) — 2 (^i^g^ + n^ gj + rf) - 3 (n^ r^ + n^i'i+x), • {B) 



where §i, $3, 1'l, i\ represent the degrees of the nodal and cuspidal 

 curves of the two surfaces S^ and S^. Formula {A) shall first be 

 proved for the case that S^ and S^ are developables. If we wish 

 to apply formula {^B) to developables the numbers of double gener- 

 ating lines tOj and to, must be added to the orders %^ and i„ of the 

 nodal curves and the numbers of stationary generating lines i\ and 

 i\ to the orders v\ and r, of the cuspidal curves. 

 Formula (5) becomes 



r = 71, n^ {n, + w, — 2) — 2,\n, (§, + (o,) + w, (|, + wj + d| — 



-3|^^,(r, + tg + '^.K + ^i)+xl . . . . (C) 



2. Let A^aS be the second polar surface of the degenerated surface 

 S, -\- S^ with respect to the arbitrary point P. This surface A*/S 

 is of the degree {n, -f- n^ — 2) and meets the curve of intersection s 

 of aSj and aSj, this curve being of the degree n^n^, in n^n^ (»i+^^j — 2) 

 points. 



These points of intersection are 1^' the triple points of /Sj-f-^^a 

 through which the curve s passes and 2"^ the points of s for which 

 the tangent plane to one of the two surfaces passes through P. 

 The triple points of S^ + ^^ through \vhich the curve s passes 

 are the points in Avhich a double line of one of the two surfaces 

 meets the other surface. So these triple points are: 



1st. Tiig {n-^v^ -|- n^r,) points in which a cuspidal curve of one of 

 the surfaces meets the other one. These points are cusps on the 

 curve of intersection s, they are indicated by Cremoxa as points X 



1) Versluys, Mémoires de Liége, 3me serie, t. VI. Sur les nombres Plückériens etc. 



2) E. Pascal, Rep. di Mat. Sup. II, p- 325. 



