1280 
binn == (ST) eeen sa oma ane (OI 
Starting from the known numbers ¢,=3, a,=1, 6,=2,¢,=1 
we can successively calculate from (4) to (7) the characteristic numbers 
for n= 2,3,4... Then it appears that the quadratic equation from 
which 4, is found always has one integer and one fraction. as roots. 
The following table is found. 
| 
n | th An by Cn 
NA mes GN ir 
91 9 6 | 2 3 
| | 
SMR SNS 7 
AD A 0 EN 
The numbers of the last column satisfy the relation 
EN Re AE (DO 
Starting from the supposition that it holds good for any n, we 
find in the first place, on account of (8) 
ll da en ee 4 te (0) 
From (6) follows 4, = en 1 + ¢,—1 consequently is ¢, = 21 + 
+ 23 or also 
NEE Un) 
From (7) follows finally 
ga A (20 Seele eo! ee A(N 
If now # = 3c, 2a, = 4c,—1-—-+(—1)" and },=c,—(— 1) 
are substituted in (4), that relation appears to be identically satisfied; 
thus the solution of the system is obtained and we have this property :*) 
The nth tangential curve of a base-point B is a curve of order 
3 (2e—1), and has in A, B, C multiple points of the orders 2 (2"—1) + 
+ + {(—4)r A), (2»—1) — (1) and (27-14). 
§ 5. The curves (4“,5B°) and 1° (A, B?,4C) have in B the 
tangents in common; of their common points 4 + 6 + 8 lie therefore 
in the base-points. The remaining three are points of inflexion and 
at the same time tangential points of B, consequently’) sewtactic 
1) For a general pencil (9%) is 
t, = 4(4"—1), 9b, = An+t 1 (—-2)p+3_5, De, = 4-41 4 (—2)"—5. 
These numbers have been found by P. H. Scuourg (CG. R. 101, p. 736). An 
extension on pencils of curves or (where each point has then (m—2) tangential 
points) is to be found in my paper “Faisceaux de courbes planes”, (Archives 
TevLer, sér. 2, tome 9, p. 99). 
2) Cf. for instance SALMON-FiepLER “Höhere ebene Kurven’’, 2d edition, p. 173, 
