h[rho] d[rho] = sin z dz, and integrating ½h[rho]² = C - cos z; (i.)
this gives the whole group of _equal-area conical projections_.
As a special case let the pole be the centre of the projected
parallels, then when
z = 0, [rho] = 0, and const = 1, we have p = 2 sin ½z/[delta]h (ii.)
Let z1 be the co-latitude of some parallel which is to be correctly
represented, then 2h sin ½z1/[delta]h = sin z1, and h = cos² ½z1;
putting this value of h in equation (ii.) the radius of any parallel
= [rho] = 2 sin ½z sec ½z1 (iii.)
This is Lambert's _conical equal-area projection with one standard
parallel_, the pole being the centre of the parallels.
If we put z1 = [theta], then h = 1, and the meridians are inclined at
their true angles, also the scale at the pole becomes correct, and
equation (iii.) becomes
[rho] = 2 sin ½z; (iv.)
this is the _zenithal equal-area projection_.