1060 
The symmetrical correspondence (MW, MW’) has as characteristic 
number  (2—5) (n—6) (10n°+4n—66) and possesses (n—5) (n-—6) 
coincidences in each of the (n—-4) (7n—9) rays WR, . The remaining 
ones arise in pairs from tangents 422,3. So we find that the inflectional 
points R, of the tangents tao3 lie on a curve of order } (n—5) (n—6) 
(13n?++457—168). 
7. Let us now consider the system {c”| of the curves which 
have the point of contact /, of their tangent t,3 on the straight 
line a. The curve (R,), cuts a in (40n—105) points R, and in 
6 (n—3)(n—4) points R,, where a osculates a c”, for which it is 
tangent t3. Consequently (R,), is of order (6n*— 2n— 33). 
The system |c"] has as index (n—4)(6n?+-15n—36)'); for (toa 
the index is, see § 4, (”—4j (9n—9). The figure produced by these 
projective systems consists of 2(n—d)(3n9) times the straight 
line a, three times the curve (R,), and the locus of the points W*, 
which each fs; has moreover in common with its c”. For the order 
of (W*), we find (7—A4) (6n?-+-15n—36) + n (n—4) (9n—9) — 2 (n—4) 
(Bn4-9) — 3 (6n?—2n—33) or (n—3) (L5n?—3n—68). 
The number of the intersections of (MW) with a again produces 
the order of the curve (/?3)33. 
The correspondence (MR,, MW) has as characteristic numbers 
(n—D5) (15n? —38n—63) and (n —5) (6n?—2n—3833), while each of the 
rays f23 passing through J/ represents (n—5) coincidences. From 
(n—5) [(15n?-—3n—63) + (6n?—2n—33) — (n—A4) (Yn—9)] we find 
now that the points of contact R, of the tangents to, are situated 
on a curve of order (n—5) (12n?+40n—132). 
The symmetrical correspondence (MW, MW,*) furnishes in the 
same way from (n—5) (n—6) | (80n?-—6n—1 26) — (n — 4) (9n—9)] the 
result, that the points of contact R, of the tangents t93 lie on a 
curve of order (n—5) n—6) (21n?+39n—162). 
8. Let us now consider the system [c”| of the curves with triple 
tangent of which one of the points of contact, #,, lies on the 
straight line a. The other two points of contact 7,, lie on a curve 
(7.)., which has two groups of points in common with a. The 
former contains the (n—5) (4n?+46n—138) points ft, of tangents 
tio (§ 6), the latter the groups of three points of contact 7, lying 
on the curves c”, for which a is triple tangent; these points are 
apparently to be counted twice. Consequently (7), is of order (n—5) 
(4n?+-46n— 138) + 8 (n—5S) (n—A4) (n-—3) or (n—5 (12n?—10n—42). 
IN. p. 940. 
