MATHEMATICS: L. E. DICKSON 
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modulo 2. For the theory^ of quadratic loci modulo 2 in space of any 
number of dimensions, see the Madison Colloquium Lectures, p. 65. 
The next case, n = 4, for which apices occur presents other remarkable 
peculiarities.^ Whereas in the algebraic case a quartic curve has 28 
bi tangents in general, one in the case of modulus 2 has at most 7 bi tang- 
ents and usually exactly 7. The bitangents intersect at derived points 
and usually all of the derived points are intersections of bitangents. 
An interesting example is given by 
K = + y' + xY + oc^z^ + y^z^ + xyz {x y + z), 
an invariant under all real linear transformations. Here real is used in 
the sense of integral; Hkewise for the real points (1, 0, 0), . . . , (1, 1,1) 
modulo 2. The bitangents to K = 0 are the 7 real lines in the plane 
and their interesections are the 7 real points, the latter being apices and 
not singular points. 
A quartic curve containing all seven real points and having no linear 
factor modulo 2 can be transformed into 
x^y + x^ + xz^ + xh"^ + yh yz^ = 0. 
It has no singular point and has the 7 apices {l,z^,z), where 2^ + 2^+1 
= 0. Its 7 bitangents are x = {b^ -\- 1) y -{- bz, where b^ -\- b -\- 1 = 0; 
they intersect at apices. Each apex is on three bitangents, while three 
apices are on each bi tangent. The configuration of the apices and 
bitangents, here all imaginary, is entirely similar to that for K, com- 
posed of real elements. 
The classification of quartic curves is similar to that next illustrated 
for the simpler case of cubic^ curves modulo 2. 
A cubic curve containing all seven real points is of the form 
a {x^y + xy^) + b (x'^z + xz^) + c (y^z + yz^) = 0. 
If not zero identically, it can be transformed into x^y + xy^ = 0. A 
cubic curve containing just two real points can be transformed into one 
containing (1, 0, 0) and (0, 1, 0); the transformations leaving the latter 
fixed or permuting them are available for the specialization of the par- 
ameters in the coefficients of the cubic. In this way we find the 21 types 
of non-equivalent cubics, including degenerate curves, and see that they 
are completely characterized by the number of real points, real inflexion 
points, real and imaginary singular points, — geometrical invariants 
easity expressed by modular invariants. 
For the determination of the inflexion points of the cubic n {xi, X2, Xs) = 
0 modulo 2, the Hessian of n is not available, being identically zero 
modulo 2. In its place we may employ the function 
