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line a 2(n-+4) (n——3)-times, because each of its points is point of 
contact of (n--4) (n—8) bitangents. The curve (/’)a belongs moreover 
twice to it. For the order of the curve (W), we find consequently 
(n —8)(7n? +-n-—6) — Un — 8)(n4+4) —2(4n?—2n—9) = (n —4)(7n?—2n — 15). 
We now consider the correspondence between the rays7’ = Mh’ 
and w= MW. A ray r’ contains (4n?—2n—9) points /’, consequently 
determines (4n?—2n—9) (n—4) rays w; to a ray w (n—4)(7n?—2n—15) 
rays 7’ are associated. Each of the (n—3)(5n—4) lines d, which 
connect J/ with the intersections of a and (f)j;, is apparently an 
(n—-4)-fold coincidence. The number of coincidences R’= W amounts 
therefore to (n—4) [ (4n? —2n—9) + (7n?—2n—-15) — (n—8) (5n—4) |= 
(n—4) (6n?+15n—36). This number is the order of the locus (R)23 
of the points of contact R of the tangents to. 
9. In order to find also the order of the locus (/)23 of the 
inflectional points 1 of the tangents f23, we return to the system [e"| 
considered in §4, of which all the curves have an inflectional point 
I on a given line a. The points S, which the corresponding stationary 
tangent has moreover in common with ec”, lie on a curve (Sao 
order 3(n?+n—5). We consider now the correspondence between 
two points S,,S, of the same curve. It determines in a pencil of 
rays (M) a symmetrical correspondence with characteristic number 
3 (n?-+n—5) (n—4). The rays connecting M with the intersections 
of a and (/)y, are (n—8) (n—4)-fold coincidences ; as their number 
amounts to 3(m—1) ($1), we find for the number of coincidences 
S,=S, [2 (nv? + n—5) — (n—1) (n—-3)] or 3 (n—A4) (n° + 6n—13). 
This, however, is also the number of tangents f23, the point of 
inflection of which lies on a, consequentiy the order of the locus 
In3 of the points of inflection of the tangents te3. 
By means of the curves (L)23 and (/)23, belonging to the system 
[ts], we can again determine the number of /ivepount-tangents t,. 
For this purpose we associate the lines MR and MJ, on account 
of which a correspondence with characteristic numbers 5 (2—4) 
(2n? +5n—12) and 3 (n—A4) (n’?+6n—13) arises. The In (n—8) (n— 4) 
tangents f 3 converging in J/ are coincidences. On the remaining 
ones A coincides with /. So we find for the number of the t, 
3 (n—A) (3n?+1412n—25) — In (n—3) (n—A4) or 15 (n—4, (4n—5S). 
10. We return to the system [c”] of the curves, which (§ 8) are 
each touched by one of their bitangents d in a point / of a straight 
line @. 
If on a line d two of the points W coincide d-becomes a triple 
