( 499 ) 



the formulae of Cayley-Plücker and for the genus is equivalent to 

 the following numbers of ordinary singularities : 



n — 1 cusps /?, 

 (n — l)(7i-hr— 3) 



2 

 m — 1 stationary planes «, 

 (w — 1) {m-\-r — 3) 



2 



r — 1 Stationary tangents <9, 

 (,._1) (,.-j-,^_3) 



2 



(r— l)(r + w-3) 



nodes H, 



T planes 



double planes G, {B) 



tangents 6, 

 double generatrices io^, 



double tana-ents a>,. 

 For a curve with only ordinary singularities we alwaj^s have 



If the curve admits of higher singularities, then the tangents in 

 these singular points will not have to count for as many double 

 tangents to the carve as they must count for double generatrices of 

 the developable belonging to the curve. The number to will then be 

 different for the formulae of Cayley-Pltjcker, relating to a section 

 and for those formulae relating to a projection, i. o. w. the singularity 

 to of a twisted curve appearing in a term {x -}- w) is not always the 

 same as the one appearing in the term (y -j- tu). 



So the formula 



is no longer correct as soon as the curve has higher singularities 

 for which order and class are unequal. 



The above as well as the following results do not hold for a 

 common cusp ^(2, 1, 1) and for a common stationary plane a (1, 1, 2), 

 the conditions {A) not being satisfied for these cyclic points. 



Through the singular point M {n, r, m) pass 



n {n-\-2r-\-m, — 4) 

 ~^ 

 branches of the nodal curve of the developable belonging to the 

 curve C. 



Ail these branches touch the curve C in M and have in M with 



the common tangent 



(n4-r)(n + 2r+m— 4) 



2 

 coinciding points in common. 



1) Salmon. 3 Dim. § 327. 



