( 627 ) 



the nodal curve on () l)e ^. Let the nninher of stationary tangents 

 he 6 and tliat of the double tangents m. Let I* he an ordinary point 

 on (', / the tangent in F to (', V the osenlating plane in I* 

 and (I the curve (jf intersection of T with the surface (J. Let c^ 

 he a conic having at P live successive i)oints in connnon with <L 

 TiCt 7M)e the vei'tex of a quadratic cone A" passing through r„ ^vhere 

 2' is arbitrary but taken in such a way that K cuts Ihe curve Cm 

 none of its singular points. Let x be the curve of intersection of (> 

 with A'. This curve, of order 2 (>, shows singular points: P^ in [* ; 

 2'"' where (' or a line ó» cuts the cone K (point I* excepted), tiiese 

 are cus[)S ). on s-, 3"^ where K meets the cui've 5 or a line oj; these 

 points are on the sui-tace 0' consisting of (J and K ti-iple ))oints 

 T a]id JU)dal j)oijits on s. 



If we now determine the points of ijitersection of s with the 

 second polar surface L'^ (>' of a jioint .1 lying in V then according 

 to Crkmona the cusps ;. must count three times and the trijde 

 points T, through \vliich oidy two branches of .v pass, must count 

 twice, in ordei' to liud out for how many points of intersection the 

 point P must count, we consider the first polar ciu-ve of the section 

 of ( >' with I^. This section consists of the right line /counted twice, 

 of the curve d and of the conic c.,. So the tirst polar curve consists 

 of the right line / and of a ciii-ve of which one i)ranch lias live 

 points in common at P with c.^ or with d ajid of a branch touching 

 / also in P and lying at P between / and d. So the second polar 

 curve shows at P two branches touching c.^ The two branches 

 of s passing through P touch / in P, and [" being also of l)Oth 

 l)ranches the plane of osculation, the}' have both with I" or with 

 ('j three points in common. Each bi-anch of .v through P has thus 

 with L^ 0' in P but four points in common, so that the point P 

 counts lor eight points of intersection. 



Besides meeting the curve .v in these singular }>oints of .v the polar 

 surface L' ()' meets it moreovei- in the ordinary points whei-e .s' meets 

 the curve of contact of tangent planes through A. These points 

 count according to Ckkmona each one time. The curve of contact 

 consists of two genei*ating lines of K each meeting the curve .s' 

 Q times and of (jt — i) generathig lines of (J, as .1 is situated in the 

 osculatinii' plane V. These lienei'ating lines each meet s two times, 

 so that to determine /. we arrive at the formula: 



2 (>' := 3 ;. 4- 4 ^ -)- 4 w 4- 8 + 2 9 -f 2 (fi - 1), 



'6X—'2\t/ — Q — ii — 2g — -lto — 3|. 



As accordinn" to one of the formulae of C.wlky-Plückkk 



