cos A = tanh (c/2γ)
(6).
The angle A is called by N.I. Lobatchewsky the “angle of parallelism.”
The whole theory of lines and planes at right angles to each other is simply the theory of conjugate elements with respect to the absolute, where ideal lines and planes are introduced.
Thus if l and l′ be any two conjugate lines with respect to the absolute (of which one of the two must be improper, say l′), then any plane through l′ and containing proper points is perpendicular to l. Also if p is any plane containing proper points, and P is its pole, which is necessarily improper, then the lines through P are the normals to P. The equation of the sphere, centre (x1, y1, z1, w1) and radius ρ, is
(w1² − x1² − y1² − z1²) (w² − x² − y² − z²) cosh² (ρ/γ) = (w1w − x1x − y1y − z1z)²
(7).
The equation of the surface of equal distance (σ) from the plane lx + my + nz + rw = 0 is
(l² + m² + n² − r²) (w² − x² − y² − z²) sinh² (σ/γ) = (rw + lx + my + nz)²
(8).