1275 
(number of the #9), consequently is of order (4n°+2 5). 
The process (a) again produces now from 2 (4n°+ 2 n—2) ) (n— 5) 
—4 (n + 3)(n—4) (n—5) the number of te4 with point of contact B, 
found above. 
The loeus (S) of the points S lying on the tangents f22>, passes 
bn? +1 7n-+56)(n—5)(n—6) times through B, is therefore of order 
4(5n? +13n-+8)(n—5)(n—6). 
If the process (a) is applied to the pairs .S,S’, we find from 
(5n?+138n+8) (n—d) (n—6) (n—7) — 2 (n + 3) (n—A4)(n—5S) (n— 6) 
(n—7) that + (8n?+15n-+32) (n —5) (n—6) (n—-7) tangents ta, have 
a point of rate in B: 5 
Extension of these considerations on the cone of the tangents 4, 
with point of contact R,, which tangents pass through 4, and on 
the corresponding loei (f#,) and (S) makes clear that through B 
(LOn? -10n-+-55) (n—5) tangents t, may be drawn, of which the point 
of contact R, does not lie in B, and (L4n?—2n-+-64) (n—5)(n—-6) 
tangents t,,,, with points of contact R,, R,. 
Analogously the cone of the tangents ¢,,,, passing through £ 
produces the number of (lan H3n4-63) (n—5) (n—6) of the straight 
lines t,,, and the number of + (A9n? ene 5) (n—6) (n—7) 
of the straight lines 
contact there. 
We finally find that through 5 4(12n + 2 5) (n—6) (n—7) 
(n— 8) tangents tf, may be drawn, of a none of the points 
of contact lie in B. 
basse passing through B, ee no point of 
§ 8. The net |] is intersected by an arbitrary plane along a 
net of plane curves gv. Making use of the results at which I have 
arrived elsewhere for a similar net‘), the order of the cone formed 
by the tangents ¢, coinciding in an arbitrary point P, may be 
determined. 
Then it will be evident, that the tangents t, form a complex of 
order 6n (n—3). 
For a base-point 5 the cone of the complex degenerates into the 
cone of order (6n-+6)(7—-4) on whose generatrices the point of con- 
tact R, does not lie in B and the cone (¢,)° 
which have their point of contact in 4; the latter is to be counted 
four times. 
The tangents tss form a complex of order In(n—8)(n—4). The 
cone of the complex of B consists of three parts: a cone of order 
of the straight lines f,, 
1) See my communication on ‘Characteristic numbers for nets of algebraic 
curves)’, (Proceedings XVII, p. 935). 
