805 
which envelope three tangents can be drawn out of QO; in other 
words three points P lie on A,A,, so that the tangent at P to the 
conic (P, A,, A,, A,, Q) passes through O, and thus we have arrived 
at the result already found in § 4. 
7. We shall now try to find the locus of the points S for which 
one of the nodal tangents to the curve of S, which has a double 
point in S, passes through the fixed point O. 
A point P of a straight line / is a double point of one curve of 
S,; this curve cuts / in one more point P’. Inversely P’ defines a 
net of cubics out of S, with six base points(A,,...,A,, P’). The 
locus of the double points of the curves of this net is of the 6*% 
order with double points in A,,..., A, and P’; it cuts / therefore 
besides in /’ in 4+ more points. Between P and P’ there exists a 
correspondence (1,4) with 5 coincidences, i.e. on any straight line | 
lie five points P, so that one of the nodal tangents of the curve 
which has a double point in P, coincides with 1. 
We can deduce from this that the envelope of the nodal tangents 
of those curves in S, which have double points in the points of the 
straight line /, is of the 7' class. 
For this reason 7 tangents can be drawn out of the point O to 
this curve belonging to’ /, so that it appears that there lie seven 
points on / where one of the nodal tangents is a straight line that 
can be considered as a part of a degeneration. 
However it is clear that also the two points of intersection of / 
with the conic (A,,..., A,) must be reckoned among these 7 points, 
so that the result is: 
The points that are double points of curves of S, where one of 
the nodal tangents is a part of a degeneration, lie on a curve of 
the 5th order. 
It is already known from § 5 that the points of this curve c, correspond 
to the points of inflexion of that system of plane cubics represented 
by S, on the surface of the 4" order ®, that corresponds to the 
straight lines through OQ. Each of these cubics has three points of 
inflexion, so that each straight line through O can cut the c, in 3 
points. The point OU is a double point of ¢,, the nodal tangents are 
the tangents at OQ to that curve of S, which has a double point in O. 
It appears further that the base points A,,...., A, are points of 
inflexion of c,; the tangents at the points of inflexion pass all through 0. 
Besides these, 4 single tangents can be drawn out of O to c,, 
namely the lines joining O to the four points A; corresponding to 
the pinch points of the represented surface ®,. 
52* 
