If it were possible to make a perfect representation, then we should have [sigma] = 1, [sigma]´ = 1 throughout. This, however, is impossible. We may make [sigma] = 1 throughout by taking [rho] = z. This is the _Equidistant Projection_ just described, a very simple and effective method of representation. Or we may make [sigma]´= 1 throughout. This gives [rho] = 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 [sigma][sigma]´ = 1, or [rho]d[rho] = sin z dz, the integral of which is [rho] = 2 sin ½z, 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 [sigma] = [sigma]´, or d[rho]/[rho] = dz/sin z, the integral of which is, c being an arbitrary constant, [rho] = c tan ½z. 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, [sigma] = ½c sec² ½z, 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 [sigma] = 1 and [sigma]´ = 1, so as to have a perfect picture of the spherical surface, yet considering [sigma] - 1 and [sigma]´ - 1 as the local errors of the representation, we may make ([sigma] - 1)² + ([sigma]´ - 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 [mu], and then integrate from the centre to the (circular) limits of the map. Let [beta] be the spherical radius of the segment to be represented, then the total misrepresentation is to be taken as _ _ _ / [beta] | /d[rho] \² /[rho] \² | | | ( ------ - 1 ) + ( ----- - 1 ) | sin z dz, _/ 0 |_ \ dz / \sin z / _| which is to be made a minimum. Putting [rho] = z + y, and giving to y only a variation subject to the condition [delta]y = 0 when z = 0, the equations of solution--using the ordinary notation of the calculus of variations--are d(P) N - ---- = 0; P[beta] = 0, dz