1058 
(n—4) (n--5) points of the curve (W)p. The ec” passing through P 
form a net, of which the tangents f23 envelop a curve of class 
In (n—3) (n—A4). *) 
As P is point of contact A, for (n—4)(8n-+-9) and point of 
contact R, for (n — 4) (n + 9) curves of the net, P lies, as point 
W, on Ins (n — 4) (Bn + 9) — 3 (n — 4) (n + 9) 
— 9 (n— 4) (n—S) (n+1) tangents ‘23. The order of (W)p amounts 
therefore to 6 (n—3) (n—A4) (n—5) + 9 (n—A4) (n—5)(n +1) or 
3 (n—A) (n—5S) (Sn—3). 
Starting from the correspondence (f,, IV) we arrive again at the 
class of the curve enveloped by 442 ($ 1). 
A new result is produced by the correspondence of the rays 
MR,, MW. Its characteristic numbers are (f (72-4) (n—ò) and 
(15n—9) (n—4)(n—5). The ray MP adr 6 (n—3) (n—4) (n—5) 
coincidences. The remaining 187 (n— 4) (n—5) arise from coincidences 
R,= W. consequently from tangents f5,5. As each /33 determines 
two coincidences, the twice osculating tangents ts,5 envelop a curve 
of class In (n—4) (n—S). come 
The symmetrical correspondence between the rays connecting J/ 
with the pairs of points IV, W’, belonging to the same c”, has as 
characteristic number (n—4) (2—5) (n—6) (15n—9). As MP represents 
6 (n—-3) (n—4) (n—5) (n—6) coincidences, and the remaining ones 
arise in pairs from tangents fo23, the tangents 1253 envelop a curve 
of class 12n (n—A) (n 
5) (n—6). 
5. The /,°, which IP determines on /, contains 4(n—3)(n— 4)\(n—5) 
groups with three double elements; as many curves c” have / as 
triple tangent f22. In the net determined by P occur 2 (n + 3) : 
(n—4) (n—5) Cc”, on which P is point of contact of a triple tangent”) 
If / rotates round P, the points of contact describe therefore a curve 
of order 4 (n—38) (n—4) (n—5) + 2 (nr + 3) (n—4) (n—5) or 6 (n—4) 
(n—-5) (n—1). 
We further determine the order of the locus of the groups of 
(n—6) points Q, which / has moreover in common with the cr, 
which it touches three times. The 422 belonging to the net with 
base-point P envelop a curve of class 2n (n—3) (n—A4) (n—5).*) As 
P is point of contact for 2 (n+3) (n—4) (n—5) c”, the number of c* 
intersecting their f22 in P amounts to 2n (n—3) (n—4) (n—5) — 
— + (n+3) (n—4) (n—5) or 2 (nd) (n—-+) (n—5) (n —6). The order 
1 
ES oF p. 936. 
2) N. p. 943. 
en BN. p. 280. 
