Thus points on the surface corresponding to points in the plane on the limiting circle r = a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle μ at the origin is
σ = Rrμ / √ (a² − r²) = μR sin h (ρ/R),
whence the circumference of a circle of radius ρ is 2πR sin h (ρ/R). Again, if α be the angle between any two geodesics
V − v = m (U − u), V − v = n (U − u),
then
tan α = a (n − m)w / {(1 + mn)a² − (v − mu) (v − nu)}.
Thus α is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v = 0 at a distance δ from the origin is u = a tan h (δ/R), and the parallel to this through the origin is u = v sin h (δ/R). Hence Π (δ), the angle which this parallel makes with v = 0, is given by
tan Π(δ) . sin h (δ/R) = 1, or tan ½Π(δ) = e−δ/R
which is Lobatchewsky’s formula. We also obtain easily for the area of a triangle the formula R²(π − A − B − C).
Beltrami’s treatment connects two curves which, in the earlier treatment, had no connexion. These are limit-lines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius ρ and centre u0v0 is