MATHEMATICS: H. S. WHITE 
337 
A VARIABLE SYSTEM OF SEVENS ON TWO TWISTED 
CUBIC CURVES 
By H. S. White 
DEPARTMENT OF MATHEMATICS, VASSAR COLLEGE 
Received by the Academy. May 15, 1916 
Seven points chosen at random on a twisted cubic curve, like six 
points on a conic in the plane, give rise to a distinctive theorem; for 
as five points determine a conic, so the twisted cubic is determined 
by six points. In the case of the conic, this is the theorem of the Pas- 
cal hexagon, six points in a definite order leading to a definite line. 
Conic and fine remaining fixed, the hexagon may vary with four de- 
grees of freedom. In the case of the twisted cubic, not a mere sequence 
of the seven points, but an arrangement of them in seven triads, deter- 
mines seven planes, and the theorem states that these planes are all 
osculated by a second twisted cubic curve. So much was estabhshed 
by a direct proof in these Proceedings in August, 1915; but the ques- 
tion of variability, whether the points and planes are free to move while 
the two curves remain fixed, was not examined. Now it is found that 
the system is variable with one degree of freedom. Full proof is contained 
in a paper soon to appear in the Transactions of the American Mathe- 
matical Society. The following is an outline. 
Every twisted cubic C3 is a rational curve, and the homogeneous 
coordinates of its points are cubic functions of a variable parameter: 
Xi = fi (X), X2 = /2 (X), . . . ,X4 =^ f^ (X). 
In the same way the osculating planes of any second cubic curve 
are represented by cubic functions of a second parameter: 
ui = gi (m), U2 = go (m), . . . ,th = gA (m). 
Every point of the first may be put in relation to the three planes 
of the other that pass through it by the equation 
Ui Xi + U2 X2 th X3 U^Xi = 0, 
or 
of the third degree in X and also in ^t. Conversely, every bicubical 
relation, or (3,3) correspondence, may be interpreted as such a point- 
to-plane relation between points of an arbitrary cubic C3 and osculat- 
