( 500 ) 



These branches have in M the same osculating plane as C and 

 with this osculating- plane they have in M 



{n-\-r-\-m) {n-\-2r-\-in — 4) 

 __ 



coinciding points in common. 



From the conditions {A) ensues that {n-\-1r-\-m) is even, so that 

 the three above nnmbers are integers. 



The second polar surface of according to an arbitrary point 

 meets in the point M{n,r,m) the cuspidal curve 



times and the nodal curve 



n-\~2r-\-m — 4 



^ (n-|-r— 2)(n+r-fm) 



times. 



Each point R, where the tangent in M still meets a sheet of 

 the surface 0, counts for 



r'-|-rm — m — r 



points of intersection of the nodal curve with the second polar surface. 

 In the equation of Cremona ^) serving to determine A (number of 

 cusps of the nodal curve) we must add for every singular point 

 M {n, r, m) in the second member of the equation a term 



{7i-\-r — 2) {?i-\-r-\-m). 



In the equation of Cremona "), serving to determine t (number of 

 triple points of the nodal curve) we must add for every singular 

 point to the second member of the equation a term 



n-\-2r-\-m — 4 



{n-\-r — 2) {n-\-r-\-m) 



Li 



and for the corresponding points U a term 



(w4-2r + m-4) {r^m) (r— I). 



The decrease of A and t arising from the presence of a point 

 M (71, r, m) is not equal to the decrease of A and t caused by the 

 ordinary singularities necessary to form a singularitj^ J/(??,, r, ?/?). So 

 the equivalence of the values expressed in {B) does not extend to 

 numbers which are found hj means of a second polar surface. 



Delft, November 1905. 



1) Gremona-Curtze, Oberflachen § 104. 

 3) loc. cit. § 109. 



