[sigma] = d[rho]/dz; [sigma]´ = [rho]/sin z,
then p´q´ = [sigma]pq and p´r´ = [sigma]´pr. That is to say, [sigma],
[sigma]´ may be regarded as the relative scales, at co-latitude z, of
the representation, [sigma] applying to meridional measurements,
[sigma]´ to measurements perpendicular to the meridian. A small square
situated in co-latitude z, having one side in the direction of the
meridian--the length of its side being i--is represented by a
rectangle whose sides are i[sigma] and i[sigma]´; its area
consequently is i²[sigma][sigma]´.
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