( 53 ) 



and must count accordiiig' to liim for three points of intersection of 

 the nodal curve, thus here of the curve s witli A'aS ^), 



2"<^. The ("i^'2+^^;^"i) points in wliich a stationary generating line 

 of one of the surfaces meets the other one. These points, also cusps 

 on the curve s, are indicated In' Cremona as points r which must 

 count according to him for tiiree points of intersection of the nodal 

 curve 6' with A'S^). 



S^'^l The (/?i^2 + "ü^i) intersections of >S'i or >S^ with the nodal 

 curve of the other surface. According to Cremona each of the branches 

 of the nodal curve meets A^*S ^) one time in a triple point t. Through 

 each of these points t pass two branches of s, which is a nodal 

 curve on S^ -{- aS'.^ ; so each of these triple points counts for two 

 points of intersection of s with A^»S. 



4t''. The {n^co^ -f '^2^'^i) nodes of s in which a double generator 

 of one of the surfaces S^ or S^ meets the other one. According to 

 Cremona the nodal curx-e .? meets A^*S *) two times in such a 

 triple point r. 



The surface >Si -\- S.^ possesses still more triple points, among others 

 the cusps I? of the cuspidal curves ; these points do lie on A^S, but on 

 the curve of intersection 0' they do not; so they do not belong to 

 the points of intersection of .s^ with A^/S. 



3. Through P pass m^ tangent planes of the surface S^. A gene- 

 rator of aS'i, along which one of the 7??i tangent planes through F 

 touches >Si, meets n^ times the surface S^. Each of these points of 

 intersection is a point on s also situated on A^S. Such a point of 

 s and of A^aS counts for one point of intersection, according to 

 Cremona ^). So A'^S is met by the curve s in {ni^n^ -f- m^n^) points 

 for which one of the tangent planes passes through P. 



This gives the relation : 



n^n, {n^ + «^ ~ ^) = ^^h^^-z + '>^h^h + '^ t^i (^3 + ^J + n^ (§, + cOj)] -|- 

 + 3 K (r, + ^) -h /^ O', + t^JI {D) 



Comparing the equations {C) and {D) Ave get immediately 



r = m^n^ 4~ ^'^3'*: — 2(f — 3/ {^A) 



The degree of a developable being the rank of its cuspidal curve, 

 we can write for this formula: 



r = m^i\ -f- m^r^ — 26 — 3/. 



1) Cremo]na — CuRTZE, Oberfliichpn § 108. 



2) Cremona — Gurtze, loc. cit. § 100. 



3) Cremona — Curtze, loc. cit. § 109. 



4) Cremona — Curtze, loc. cit. § 101. 

 ») Cremona — Curtze, loc. cit. § 99. 



