(714) 
through B bears but (2n—4) points 7, whilst the curve ¢ is of 
order (22 — 1). 
Of the common points of "—! and s*"@—®*) there are 3 (n + 4) (n — 3) 
lying in B, 2 (2 — 3) in each of the remaining (n° — 1) base-points and 
two in each of the inflectional points sending their tangent through B. 
The number of those inflectional tangents is 3” (n — 2) —9, as 
each of the three inflectional tangents, having their inflectional point 
in B, must be counted three times. This is evident when we consider 
a curve of (c*), where a base-point can lie only on inflectional 
tangents for which it is inflectional point itself. This number amounts 
to three, whilst the class of the envelope of the inflectional tangents 
is nine. 
So we find for the number of the points of contact, not lying in 
B, of double tangents out of B 
37(n—8) (2n—1) —3 (nd) (n—3) —2 (n—8) (n?>—1) —6 (n—8) (n41) = 
=4(n—3)nm—J4An+1). 
So B lies on 2 (n — 4) (n — 3) (n + 1) double tangents. This num- 
ber is 2 ( 3) (n +4) less than the number of double tangents out 
of an arbitrary point. The ( 3) (n+ 4) double tangents having 
one of its points of contact in B must thus be counted twice. 
The envelope of the double tangents has in each base-point an 
(n + 4) (n — 3)-fold point. 
§ 4. The locus of the points of contact D of the double tangents 
of (ct) evidently passes (n + 4) (7 — 3)-times through each base-point 
($ 3). An arbitrary c” having on its double tangents 7 (7% — 2) (n? — 9) 
points of contact D, the curve D and c” intersect each other in 
n° (n + 4) (n — 8) + n(n — 2) (n° —-9) points. Consequently the locus 
of the points of contact D is a curve of order (n—8)(2n?-+-5n—6). ') 
We shall now consider the locus of the points IV in which a 
cr is intersected by its double tangents. 
As each base-point B lies on 2 (m — 4) (n 3) 2 + 1) double tan- 
gents ($ 3) the curve W passes with as many branches ith B. 
So it has with an arbitrary c* in common 22° (n— 4) (n—9) (n-+1) + 
+ 42 (n — 2) (n? — 9) (n —4) points. From this ensues that the curve 
(W) is of order 4 (72 — 4) (n — 3) (5n? + 5n — 6). 
The curves (D) and (W) have outside the base-points a number of 
points in common equal to 
4 (n — 4) (n — 3)? (5n? + 5n — 6) (An? + Sn — 6) — 
— 2n? (n — 4) 3)? (n + 1) (n + 4). 
1) See P. H. Scuoure, Wiskundige opgaven, II, 307. 
