The object of this paper is to give some account of the geometry of the three-dimensional space
S
wherein the co-ordinates belong to a Galois field
K
of 3 marks. A description of the fundamental properties of quadrics is sufficiently long for one paper, and so an account of the line geometry is deferred. The early paragraphs (§§ 1 to 4) are necessarily concerned with geometry on a line or in a plane. A line consists of 4 points; these are self-projective under all 4! permutations. A plane consists of 13 points and has the same number, 234, of triangles, quadrangles, quadri-laterals and non-singular conics. A diagram is helpful, especially when we consider sections by planes in
S
. The space
S
has 40 points. Non-singular quadrics are of two kinds: either ruled, when we call them hyperboloids, or non-ruled, when we call them ellipsoids. A hyperboloid
H
consists of 16 points and has a pair of reguli; the 24 points of
S
not on
H
are the vertices of 6 tetra-hedra that form two allied desmic triads. The ellipsoid
F
is introduced in § 12; it consists of 10 points, the other 30 points of
S
being separated into two batches of 15 between which there is a symmetrical (3, 3) correspondence. Either batch can be arranged as a set of 6 pentagons, each of the 15 points being the common vertex of 2 of these. The pentagons of either set have all their edges tangents of
F
and, with their polar pentahedra, have significant properties and interrelations. By no means their least important attribute is that they afford, with
F
, so apposite a domain of operation for the simple group of order 360. In §§ 23 to 26 are described the operations of the group in this setting. Thereafter the 36 separations of the 10 points of
F
into complementary pentads are discussed, no 4 of either pentad being coplanar. During the work constructions for an ellipsoid are encountered; one is in § 16, another in § 30.
On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation
