Or we may make σ′= 1 throughout. This gives ρ = sin z, a perspective projection, namely, the Orthographic.
Or we may require that areas be strictly represented in the development. This will be effected by making σσ′ = 1, or ρ dρ = sin z dz, the integral of which is ρ = 2 sin 1⁄2z, which is the Zenithal Equal-area Projection of Lambert, sometimes, though wrongly referred to as Lorgna’s Projection after Antonio Lorgna (b. 1736). In this system there is misrepresentation of form, but no misrepresentation of areas.
Or we may require a projection in which all small parts are to be represented in their true forms i.e. an orthomorphic projection. For instance, a small square on the spherical surface is to be represented as a small square in the development. This condition will be attained by making σ = σ′, or dρ/ρ = dz/sin z, the integral of which is, c being an arbitrary constant, ρ = c tan 1⁄2z. This, again, is a perspective projection, namely, the Stereographic. In this, though all small parts of the surface are represented in their correct shapes, yet, the scale varying from one part of the map to another, the whole is not a similar representation of the original. The scale, σ = 1⁄2c sec2 1⁄2z, at any point, applies to all directions round that point.
These two last projections are, as it were, at the extremes of the scale; each, perfect in its own way, is in other respects objectionable. We may avoid both extremes by the following considerations. Although we cannot make σ = 1 and σ′ = 1, so as to have a perfect picture of the spherical surface, yet considering σ − 1 and σ′ − 1 as the local errors of the representation, we may make (σ − 1)2 + (σ′ − 1)2 a minimum over the whole surface to be represented. To effect this we must multiply this expression by the element of surface to which it applies, viz. sin zd zd μ, and then integrate from the centre to the (circular) limits of the map. Let β be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as
| ∫β0 { ( | dρ | − 1 )2 + ( | ρ | − 1 )2 } sin z dz, |
| dz | sin z |
which is to be made a minimum. Putting ρ = z + y, and giving to y only a variation subject to the condition δy = 0 when z = 0, the equations of solution—using the ordinary notation of the calculus of variations—are
| N − | d(P) | = 0; Pβ = 0, |
| dz |
Pβ being the value of 2p sin z when z = β. This gives
| sin2 z | d2y | + sin z cos z | dy | − y = z − sin z ( | dy | ) β = 0. |
| dz2 | dz | dz |
This method of development is due to Sir George Airy, whose original paper—the investigation is different in form from the above, which is due to Colonel Clarke—will be found in the Philosophical Magazine for 1861. The solution of the differential equation leads to this result—