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that the four coincidences coincide two by two; for, if we call one 
of the two chords 5 through such a point 6,, then the other is 6,, 
but if we call the latter 6,*, then 6, —6,*, so that really the coin- 
cidences coincide two by two. Furthermore it is easy to point out that 
in general the two coincidences do not fall in the points of intersection 
of a and k*; for, both chords 5 through such a point will in general 
not lie with a in one plane. 
So out of these considerations follows that a intersects the demanded 
locus outside £* in two more points; if therefore we point out that 
ke is a nodal curve, then we have proved that the demanded locus is 
a surface 2° of order 6. Now through a point P of 4’ pass two 
chords 4 and in the plane through these lie two chords a; so each 
point of 4° is a nodal point for the surface. 
2. We again determine the order of 2* by considering a chord 
hb, of &*. Through a point P of 6, passes one a and in the plane 
ab, lies one b,; if the latter intersects 6, in @ then to each point 
P one point @ corresponds. Inversely through @ passes one b,, 
but in the plane 4,6, lie three chords a; so on 6, we find now a 
(1,3) correspondence with four coincidences, and these do not coincide 
two by two. For, through each coincidence passes one a and one 6, 
but of course these cannot be exchanged. Neither does a single 
coincidence fall on 4“; for through a point of intersection P of 6, 
and 4* passes one a and the line connecting the two remaining 
points of intersection of plane ab, and #* does of course in general 
not pass through P. So a chord of £* cuts 2° outside £* in four 
points more; therefore k* is for 2° a single curve. 
This last result has something unexpected, for if we regard £* by 
itself we arrive at quite a different result. Through a point P of 4% 
passes one @ and in an arbitrary plane through this lie three chords 
6 through P; so that each point of 4* regarded by itself satisfies the 
given question an infinite number of times; if however we also take 
into consideration the points outside 4°, then we find according to 
the above mentioned a surface 2° for which £‘ is only a single curve. 
That 4 is just a single curve is made clearer by the following 
consideration. The curve 4* is the section of two quadratic surfaces 
P,,®,, and the plane of the two chords 6,,6, is at the same time 
the plane through P and the line of intersection s of the two polar 
planes 2, z,, of P with respect to ®, and ®, ; if now P falls exactly 
on £*, then ar, ar, become tangential planes in P to ®,, ®,, so their 
line of intersection s becomes the tangent ¢-in P to £*; among all 
the planes through ? only those through ¢ come into consideration, 
