(a² − uu0 − vv0)² = cos h² (ρ/R) w0²w² = C²w²
(say).
This equation remains real when ρ is a pure imaginary, and remains finite when w0 = 0, provided ρ becomes infinite in such a way that w0 cos h (ρ/R) remains finite. In the latter case the equation represents a limit-line. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre u0v0. One of these, obtained by putting C = 0, is the straight line a² − uu0 − vv0 = 0. Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.)
The above Euclidean interpretation fails for three or more dimensions. In the Teoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of the Saggio)
ds² = R²x−2 (dx² + dx1² + dx2² + ... + dxn²)
where
x² + x1² + x2² + ... + xn² = a².
He shows that geodesics are represented by linear equations between x1, x2, ..., xn, and that the geodesic distance ρ between two points x and x′ is given by
| cos h | ρ | = | a² − x1x′1 − x2x′2 − ... − xnx′n |
| R | {(a² − x1² − x2² − ... − xn²) (a² − x′1² − x′2² − ... − x′n²)}1/2 |
(a formula practically identical with Cayley’s, though obtained by a very different method). In order to show that the measure of curvature is constant, we make the substitutions