| ρ = 2 cot 1⁄2z loge sec 1⁄2z + C tan 1⁄2z, |
| C = 2 cot2 1⁄2β loge sec 1⁄2β. |
The limiting radius of the map is R = 2C tan 1⁄2β. In this system, called by Sir George Airy Projection by balance of errors, the total misrepresentation is an absolute minimum. For short it may be called Airy’s Projection.
Returning to the general case where ρ is any function of z, let us consider the local misrepresentation of direction. Take any indefinitely small line, length = i, making an angle α with the meridian in co-latitude z. Its projections on a meridian and parallel are i cos α, i sin α, which in the map are represented by iσ cos α, iσ′ sin α. If then α′ be the angle in the map corresponding to α,
tan α′ = (σ′ / σ) tan α.
Put
σ′ / σ = ρ dz / sin z dρ = Σ,
and the error α′ − α of representation = ε, then
| tan ε = | (Σ − 1) tan α | . |
| 1 + Σ tan2 α |
Put Σ = cot2 ζ, then ε is a maximum when α = ζ, and the corresponding value of ε is
ε = 1⁄2π − 2ζ.