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918 
Mathematics. — “On the correspondence of the pairs of points 
separated harmonically by a twisted quartic curve.” By Prof. 
JAN DE VRIES. 
(Communicated in the meeting of November 30, 1912). 
§ 1. We indicate by P and Q two points, lying on a chord ofa 
twisted quartic curve of the first kind, separated harmonically by this 
curve of. As any point P lies generally on two chords, in the 
correspondence (P, Q) to any point P two points Q are conjugated. 
If: F moves along a line /, Q describes a curve 4° of order sw. 
For any plane 4 through / cuts 9* in four points Sj, and contains 
therefore six points Qi, where Q,; lies on a chord SS, and is 
harmonically conjugated to the points #%/ common to that chord and 
/. If / is an arbitrary line, Q never lies on / when 4 rotates about /. 
The line Q,,Q,, is separated harmonically from / by P,,S, and 
SS. By assuming a position for / in which S, and S, coincide 
with Q,, we find for Q,,Q,, a tangent of ° separated harmonically 
from / by P,,S, and P,,S,, whilst an other tangent of 2° takes the 
place of Q,,Q,,- So each of the eight tangential planes of 9 
contains two tangents of 4°; so the rank of this curve is sixteen. 
Moreover we find that 4° has eight points in common with 9%. 
§ 2. The line p connecting the two points Q, Q’ conjugated to 
P describes a regulus 4’ if P moves along /. For p is the polar 
line of P with respect to e°‚, i.e. the intersection of the polar planes 
of P with respect to any two quadratic surfaces through g*, and 
these polar planes describe two projective pencils. 
Let us now consider one of the two lines p cutting /. The 
corresponding point P bears two chords S,S, and 98. lying in the 
plane 4 = ip. The points Q,, and Q,, lie on p, the pomts Oan 
Q,,,Q,, lie on a line m through P harmonically separated from / 
by the chords S,S, and S,S,. As À° lies on the regulus 4°, m isa 
line of 4°. Any tangential plane of A’ contains therefore a quadri- 
secant of 2° and beth the reguli of 4° are arranged by A° in a 
correspondence (2,4). Evidently the quadrisecants gq are the polar 
lines of / with respect to the quadratic surfaces through g’*. 
§ 3. If we assume for / a chord of «*‚ the locus of Q breaks 
up into four parts, i.e. the chord @ itself, the tangents r and 7’ in 
the points R, R’ common to / and v‘, and a twisted cubic 1°. The 
polar line p now connects a point Q of / with the point Q’ of the 
second chord & passing through P. This line describes a regulus 
