936 
the same cr. Any ray s contains 3 (n—3) (2n—1) points S, determines 
therefore as many rays s'; any ray s contains 3 (n—1) points E 
determines therefore 8 (n—1)(n—8) points S and consequently as 
many rays s. The number of rays of coincidence s — s amounts 
therefore to 3 (n—3) (2n—1) + 3 (n—3) (n—1) = 3 (n—3) (Bn—2). The 
ray MP contains 3 (n—2) points /, which are each associated to (n— 3) 
points S; consequently MP represents 3 (n—2) (n—3) coincidences. 
The remaining 6n (n — 3) coincidences arise from coincidences [= $, 
consequently from points of undulation U. Through P pass conse- 
quently 6n(n—8) four-point tangents t,; the tangents t, envelop there- 
fore a curve of class 6n (n—3). 
3. We further consider the correspondence between the rays 
SS, Which connect M with two points S belonging to the same 
point /. This symmetrical correspondence has apparently as charac- 
teristic number 38 (2—8) (2n—1) (n—4). The ray MP contains 3 (n—2) 
points of inflection, hence 3 (n—2) (n—3) (n—4) pairs S,, S, ; as many 
coincidences s,—s, coincide with MP. The remaining coincidences 
pass through points of contact of tangents 42,3 (straight lines, which 
touch a c* in a point R and osculate it in a point /). The tangents 
log envelop therefore a curve of class 9 n(n—8s) (n—A). 
4. Let a be an arbitrary straight line; each of its points is, as 
base-point of a pencil, point of inflection for three c”. The curves 
cr coupled by this to a form a system {c"] with index 6 (n—l); 
for the inflectional points of the curves c”‚ which pass through a 
point P, lie on a curve of order 6 (n —1)*), and the latter cuts a 
in 6(n—1) points 4. The stationary tangents 2, which have their 
point of contact J on a, form a system [4] with index 3 (m — 1), 
for through a point P pass the straight lines 2, which connect P 
with the intersections of a and (/)p. 
The systems [c”| and [4] are projective; on a straight line / they 
determine between two series of points a correspondence which has 
as characteristic numbers 6(n—1) and 38(n—1)n. The coincidences 
of this correspondence lie in the points, in which / is cut by the 
loci of the points 7 and S, which every z determines on the associated 
cn. As any point of a is point of inflection for three c", a belongs 
nine times to the locus in question. Hence the points S lie on a 
curve (S)q of order 3 (n?-+-n—5). 
For n=8 the number 21 is found; this is in keeping with the 
1) T. p. 104. 
