( 716 ) 
their tangents through the point M chosen arbitrarily is a curve of 
order (4m — 5); for M is a node of a c", so it lies on two branches 
of (K). 
Each point A’ of the arbitrary right line / is a node of a curve 
belonging to (c”),. The points of intersection Mand M' of the tangents 
in AK with the right line m chosen arbitrarily are pairs of a sym- 
metric correspondence with characteristic number (47 — 5). To the 
coincidences belongs the point of intersection J/, of / and m, and 
twice even, because the c”, having in that point a node, furnishes two 
points J/,' coinciding with J/,. The remaining coincidences originate 
from tangents in cusps. From this ensues: 
The locus of the cusps of a threefold infinite linear system of 
curves of order n is a curve of order 4 (An — 3). 
Mathematics. — “Some characteristic numbers of an algebraic 
surface.’ By Prof. Jan pm Vrins. 
In tbe following paper we shall show how by easy reasoning 
we can find an amount of the characteristic numbers of a general 
surface of order 7'). To this end we shall make use of scrolls 
formed by principal tangents or double tangents. 
§ 1. First I consider the scroll A of the principal tangents a of 
which the points of contact A lie in a given plane a. The curve — 
«along which « cuts the surface og" is evidently nodal curve of 
A. The tangents in the 87 (7% — 2) inflectional points of «* being 
principal tangents of g”, the seroll A has 37 (2 — 2) right lines and 
the curve @ to be counted twice in common with 0”, so if is a 
scroll of order 7 (Bn — 4). 
The two principal tangents « and «im a point of e@ have each three 
points in common with 9g”; consequently a” belongs six times to the 
section of A and 4”. These surfaces have moreover a twisted curve 
of order 1? (82n—4)—6n in common containing the Su(n— 2) w— 3) 
points where gr is cut by the principal tangents a situated in «. In 
each of the remaining (lin — 24) points of intersection of this 
curve with « the surface o” has four coinciding points of inter- 
section in common with a. From this ensues: 
The locus of the points in which p*_ possesses a fourpoited tangent 
(fleenodal line) 7s « twisted curve of order n (1ln—24). 
1) We find the indicated numbers in Satmon-Fiepier, “Analytische Geometrie 
des Raumes”, dritte Auflage, IL, p. 622—644, and in Scuupert, “Kalkül der abzählenden 
Geometrie”’, p. 236. 
