The collineations in question fall into two classes according as a3= + Vat at or cg, = — V a, a2. The first class constitute a group, or closed system; i.e., combination of two of them is always one of the class. Thus the transforma tion xi =ai x'1, X, = al es, xi al at x's fol lowed by the transformation =-- l3, x% Pa'2, = + V /31/32x's is equivalent to the transformation xi Pi x'i, x2 P2 x"2, ft= -I- V al at pi /32 x',, of the same class ; while if the negative sign be used in the com ponent transformations, the resultant contains are V al al x 's hence the second class closed, is not a group. Geometrically the two classes are distinguished by the fact that the transformations of the first class convert into itself each of. the segments of C into which C is separated by the contact points of x2=0 and x2=0, while the transformations of the second class exchange these segments. To the collineations of the first class Klein has given the name displacements of the plane for the reason that they correspond to the collinea tions —called displacements — that in the ordi nary Euclidean plane leave the line at oo un changed. The second class correspond to col lineations in the ordinary theory that convert figures into inversely congruent figures.
We have, therefore, the proposition: The group of displacements leave fixed not only the absolute C but also all the conics (circles in this theory) tangent to C at the two fixed points. The equation of these circles is xa x9 xl-0. Through any point x'i there passes in general one and but one of these cir cles, viz., that whose equation is x'2' x, x2 — x'2 xe= 0. A displacement of the plane causes each point to move on that circle on which it chances to lie. Accordingly each displacement is a rotation of the plane about a fixed point, all other points describing concentric circles (generalized) about this point. Those dis placements in which the centre of rotation is at CO are specially noteworthy. To be at co the centre must be on C. Hence the circles of motion have 4-point contact with C at their common centre. In the ordinary, or parabolic, geometry the centre would be on the line at on and the circles would have their centre at on, i.e. they would be a system of parallel lines. Motion along these is translation. Hence translation in ordinary (parabolic) geometry corresponds to those displacements (in the gen eralized doctrine) where the centre of rotation is on the absolute conic C.
Distances (Angles) Invariant under Dig. Let xi and x'2 be any two points and let their range cut C in ui and lei. The an harmonic ratio of these points is unchanged by collineation. Now a displacement converts u and u' into points vi and v'i of C, and xi and x'i pass into yi and y'i collinear with vi and v'i. Hence distance between x and x'i = dis tance between yi and y'2. Similar argument shows that angles are unchanged by displace ments. Neither are distances and angles changed in size by the non-group class of collineations not changing C.
Generalized Measurement for Sheaf of Lines or The line geometry of the sheaf is analogous to the point geometry of the plane, and the plane geometry of the sheaf i gnite parallel to the line geometry of the plaza The four theories may be studied simul ly by means of the same algebraic machiner In order to transfer the generalized metric the plane to the sheaf, we take for absolute M the latter a cone of second order (class) haw- ' ing the carrier point P of the sheaf for ver tex. The angle of two lines L and L' of thr
sheaf will be an arbitrary constant c times elm logarithm of the anharmonic ratio of these lines and the pair of lines common to the al solute and the plane (pencil) of L and L: Reciprocally, the angle of two planes and is an arbitrary constant c times the of the anharmonic ratio of these planes the two planes tangent to the absolute ow and containing the common line of and Developments in case of the sheaf quite sim ilar to those found for the plane are obwioestc obtainable in similar manner and need not k further pursued.
The Elliptic and Hyperbolic Theories d the Returning to the plane, it is dt vious that two theories will arise acconfmt as the absolute C is real or is imaginary. Ii be imaginary, the two absolute points of e'ec real range (line) will be conjugate imaginaries These being the infinite points of the range this will have no infinite real points. The will be finite in length, the length depending the value assigned to c. No pencil will hart real tangents to C. No two real lines intersect on C, i.e., at on , hence no two tea: lines can be parallel. The resulting theory is the so-called Elliptic geometry of the plane If our plane really is elliptic instead of being, as we commonly assume, Euclidian (or para bolic) then the infinite region of it is an imaginary conic section.
If C be supposed to be real, the resulting theory accords with that of Lobachevski are, Bolyai. It is that called Hyperbolic by Mem The real points now fall into two classes; the class E of points such that from each two real tangents can be drawn to the absolute; and the class I such that no real tangents proceed frm its points to the absolute. These classes may be respectively described as exterior and in terior. Similarly real lines compose two classes: those that cut C in two real points. and those that cut it in imaginary points Suppose, as we must make a choice, we con fine ourselves to the class I and to the lines that go through the points of I. No pencil has real lines at on, i.e., no real tangents to the abso lute. The absolute lines of any pencil are imaginary (like the isotropic lines of ordinary projective geometry). Hence, as before indi cated, we take c'=--sc'i, pure imaginary. The sum of the angles of any pencil is, therefore, 2c'or. On the other hand, every real straight line has two points at oo, namely, eints th where it cuts the absolute. Accordingly we regard c as a real quantity. Owing to our choice of c and c' the angle of any two lines intersecting within the absolute is real, and real also is the distance between any two points of I. But two lines having points in I but not intersecting within the absolute make an imaginary angle with each other ; and the dis tance between two points, one in I and the other in E, is imaginary. The distance from any point to the absolute is infinite. The angle of two lines intersecting on the absolute is zero. Such lines are parallel.