(16)
and Λ the same function of X, Y, Z, so that λ, Λ play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ.
These new functions are shown to be connected by the bilinear relation
| λ = | αΛ + β | , αδ − βγ = 1, |
| γΛ + δ |
(17)
in accordance with the annexed scheme of transformation of co-ordinates—
| Ξ | Η | Ζ | |
| ξ | α2 | β2 | 2αβ |
| η | γ2 | δ2 | 2γδ |
| ζ | αγ | βδ | αδ + βγ |
where
ξ = x + yi, η = −x + yi, ζ = −z,
Ξ = X + Yi, Η = −X + Yi, Ζ = −Z;
(18)