1056 
a pencil of rays with vertex J/ by the pairs of associated points Zi, 
and S. Any ray MR, contains (4n—6) points A, consequently 
determines (dn— 6)(n— 4) points S; any ray MS contains (n—4)10n—6) 
points S, produces therefore as many rays MA. To the (n—4)(142—1 2) 
coincidences, the ray MP belongs 4 (n —3) (n—4) times; for on MP 
lie 4 (n— 8) points R,, hence 4 (n—3) (n—A4) points S. The remaining 
coincidences arise from the coinciding of a point A, with one of 
the corresponding points S. This takes place in the point of contact 
R, of a er with a five-point tangent ¢,. From this it ensues that the 
five-point tangents of T envelop a curve of class 10n (n—4). 
We further consider the symmetrical correspondence between the 
rays of (J/), containing two intersections S,S’ belonging to the same 
point of contact #,. Its characteristic number is apparently (10n—6) 
(n—4) (n—5). On MP lie 4 (n—8) (n—4) (n—S) pairs S, S’; as many 
coincidences are represented by MP. The remaining coincidences arise 
from the coinciding of a point S’ with a point S, hence arise from 
lines #42, which have with a c* ina point R, a four-point contact, and 
in a point R, a two-point contact. The tangents ts, envelop therefore 
a curve of class 16n (n—4) (n—S). 
2. Any point of the arbitrary straight line a is, as basepoint of 
a net belonging to J, point of contact A, for six curves c”. The 
sextuples in this way coupled to a form a system |{e”|, of which 
the index is equal to the order of the locus of the points of undu- 
lation Rk, on the curves of the net set apart out of by an arbitrary 
point P, consequently equal to 3(6n— 11) *). The tangents ¢,, of which 
the points of contact &, lie on a, form a system [é,| with index 
(4n—6) for through P pass (4n—6) straight lines ¢,, having their 
point of contact R, on a ($ 1). Two projective systems [cr | and [cs | 
with indices @ and o produce a curve of order (ro + gs). If to each 
cr of the above system the line ¢, is associated, which touches it on 
a, a figure arises of order 3(62n—11) + n(4n—6). The latter consists 
of the straight line « counted 24 times, and the locus of the points 
S, which each ¢, has moreover in common with the corresponding 
cr. This curve (S)a is therefore of order (4n?-+-12n—57). 
For n= 4, (S), is therefore of order 55. In a complex of curves 
ct occur therefore 55 figures consisting of a c’ and a straight line c’. 
If all cf pass through 11 fixed points the straight lines c* are appa- 
rently the sides of the complete polygon determined by the base-points. 
To the intersections of (S)a with a belong the 4(n-—3) groups of 
1) N bl. 937, 
