941 
tangent. The correspondence between two points W,, W, of asame 
c” determines in the pencil of rays 7 asymmetrical correspondence 
with characteristic number (7n? — 2n — 15) (n — 4) (n — 5). As each 
bitangent through J/ having one of its poinis of contact on a, 
represents (n— 4)(n— 5) coincidences, the number of coincidences 
W,= W, amounts to 2 (n—A4) (n—5) (7n?—2n—15) — (n—4) (n—S) 
(n—8) (5n—4) =(n—4) (n —-5) (An? + 15n—42). As they lie two by 
two on tangents to22, the locus of the points of contact of the triple 
2,2 
tangents is a curve Roos of order } (n—4) (n—5) (8n* + 5n—14). 
We consider now the system [c”] of the curves possessing a 
tangent too, and determine the order of the loeus of the points Q, 
which each cr has moreover in common with its 4,2. The system 
[c*] has as index (n — 3) (n — 4) (n — 5) (n° + 3n — 2); for this is 
the number of c* of the pencil determined by a point P possessing 
a toes ©). The index of the system [f292]| is (§7) 2n (n — 3) (n — 4) 
(n—-5). To the figure produced by |[c”] and [f22| the curve (R)oo° 
23454 
belongs twice. For the order of (Q) we find consequently (2 — 3) 
(n— 4) (n— 5) (n° + Bn — 2) + 2n? (n— 3) (n — 4) (n— 5) — 
3 (n — 4) (n — 5) (Bn? + 5n— 14) or (n — 4) (n — 5) (n — 6) (8n? + 
+ 3n — 8). 
11. On each 4292 we associate each of the points of contact. R 
to each of the intersections Q, and consider the correspondence 
(WR, MQ). Its characteristic numbers are $ (Bn? + 5n — 14) (n— 4) 
(n — 5) (n — 6) and 3 (n — 4) (n— 5) (n — 6) (Bn? +3n—8). Each of the 
2n (n—3) (n—4) (n— 5) tangents f222 converging in JZ, represents 
apparently 3 (2 — 6) coincidences. Taking this into consideration we 
find for the number of coincidences R= Q, consequently for the 
number of tangents toa, $(n — 4) (n — 5) (n — 6) (5n? + 23n — 30). 
The correspondence between two points Q belonging to the same 
ct determines in the pencil of rays (J/) asymmetrical correspondence 
with characteristic number (3n°+3n—8) (n—A4) (n—5) (n—6) (n—7). 
To this each of the 2n(n—3) (n—4) (n—5) tangents converging in J/ 
belongs (n —6) (n—7)-times. Paying attention to this we find for the 
number of coincidences Q,=Q, 4(n—4) (n—5) (n—-6) (n—7) (n—1) 
(n+-4). There are consequently (2—4) w—5) (n—6) (n—7) (n-—1) nd) 
quadruple tangents. 
12. We shall now consider the system of the curves ct possessing 
a tangent f3, which touches it in a point #, and osculates it in a 
point J. In order to find the locus of the points S, which ec” has 
Dy Pe pe AG, 
