1277 
By means of the process (a) we find now that (6n*—18n}11) 
(n--4)+(10n?—20n-+ 14)(n—-4)— 6n(n—3)(n—4) straight lines pass 
through P, which bave a five-point contact elsewhere. The order 
of the congruence |t,| amounts therefore to AOn*— 20n-+-25)(n—4). 
We found above. ($7) that the base-point B lies on (lOn?—10n 
+55)(n—5) straight lines ¢,, which have their point of contact FR, 
not in B, whereas there are 35 straight lines ¢,, on which &, coin- 
_ cides with B. From this it ensues that each of these 35 straight 
lines must be counted five times. 
The class of the congruence |t,| agrees with the number of those 
curves g” of a net, which possess a tangent ¢,'); it is therefore 
equal to 15(4n—5)(n—4). 
$ 10. Through B pass (15n?+38n-+-63) (n—5)(n—6) tangents 43,3, 
of which none of the points of contact lies in 4, and (n° +11n445) 
(n—}) straight lines 433, which osculate in B. As they must be 
counted thrice, we find for the order of the congruence [tss] the 
number of (15n'—84n?+78n—243)(n—5). The class amounts to *) 
$(n? + 7n—9)(n —4)(n—5). 
In a similar way we find, that the congruence |ty2| is of order 
(14n'—78n?+116n—112\(n—5) and of class 6(n?+11n—14)(n— 4) 
(n-—5)*), the congruence |t22,3 | of order $(19n°>—99n?+-122n— 120) 
(n—5)(n—6) and of class $(5n?-+23n— 30)(n—4)(n— 5)(n—-6)*), the 
congruence |t2229| of order %(18n?—44n—1 12)(n—5)(n—6)(n—7) and 
of class (n—1)(n+4)(n— 4)(n—5)(n—6)(n —7)?*). 
Mathematics. — “Tangential curves of a pencil of rational cubics’’. 
By Professor Jan pr Vrins. 
(Communicated in the meeting of January 29, 1916). 
$ 1.. We consider a pencil (p°) of rational cubies which all have 
a node in A; each of the remaining base-points will be indicated 
by B or by C. 
The tangent 6 in B at ~ intersects the latter in the tangential 
point B’; the locus + of the points B’ is called tangential curve of 
B. It is generated by the projective pencils (p°) and (4). 
One of the figures of (y*) consists of the straight line AB and 
1) See the communication quoted above in volume XVII of these Proceedings 
(p. 938). 
2) Le. p. 942. 
5) Lc. p. 941. 
