1059 
of (Q) is therefore equal to 2 (n+-1) (n—4) (n—5) (n-—6) + 4 (n—3) 
(nd) (n—5) (n—6) or } (5n—3) (n—4) (n—5) (n—6). 
The correspondence (MR, MQ) produces again the class of the 
envelop of 4223 ($ 4). 
From the symmetrical correspondence (MQ, MQ), which has as 
characteristic number 3 (5n—8) (n— 4) (n—5) (n—6) (n— 7) and has in 
MP + (n—3) (n—4) (n—5) (n—6) (n—7) coincidences, we find that the 
quadiuple tangents tsooa envelop a curve of class 4 n(n—A4) (n— 5) 
(n—6) (n—7). 
6. Any point of the arbitrary straight line a, is, as base-point 
of a net, point of contact A, of (n—-4) (n+9) tangents f25.1) The 
locus (,), of the corresponding points of contact PR, has two groups 
of points in common with a; the first group contains the (407 — 105) 
intersections with the curve (#,), the second contains the 6 (n—3) 
(n—4) points A, where a is touched by the curves c”, osculating 
it in a point A, From this ensues that (2,), is of order (62?—2n—33). 
In order to find the order of the locus of the points JV’, which 
each fo3 has in common with its c”, we consider the figure produced 
by projective association of the corresponding systems [c”] and [to |. 
The curves c”, of the*net determined by P, which possess a fos, 
have their points of contact R, on a curve of order 3 (n—4) 
(n?+6n—13)*); the latter intersects a in the points /, of the curves 
of [c"] passing through P. The index of [f3] is, see § 4, (n—4) 
(7n—9). Considering that the figure produced is composed of 3 (n—4) 
(n+9) times the straight line a, twice the curve (R,)a and the 
locus (W),, we find for the order of tbe last-mentioned curve 
(n—A4) (Bn* + 18n—39) + n (n—4) (7n—9) — 3 (n—A) (n + 9) — 
— 2 (6n?—2n—33) = (n— 5) (10n? + 4n—-66). 
The curve (W), cuts a in 6 (n—3) (n—4) groups of (n—5) points 
W; in each of the remaining intersections a c” has a four-point 
contact with a line 449. Consequently the points of contact R, of the 
tangents ty. lie on a curve of order (n—5) (4n?+46n—138). 
The pairs of points R,,W determine in a pencil of rays (M) 
a correspondence with characteristic numbers (n—5) (10n’?+-4n—66) 
and (n—®5) (6n?—2n—33). The (n—4) (7n—9) rays to passing through 
M, which have their point of contact AR, on a, represent each 
(n—5) coincidences. From this ensues that the points of contact 
(inflectional points) of the twice osculating lines are situated on a 
curve of order (n—5) (9n?+39n—135). 
1) N. p. 942. 
8) N. p. 940. 
10* 
