As we saw in the discussion of the McCaw Modified Rectangular Polyconic projection, the scale transverse to a parallel is equal to:
1 1
( --------------- + 1 ) cos(phi) + ---------------
2 2
( sin(lat) ) ( sin(lat) )
To make the projection conformal for one parallel, we integrated the reciprocal of the scale along the parallel. To make it equal-area, we must instead integrate the scale as a function of phi to obtain the longitude.
The integral of the cosine is the sine, making the integral trivial, so we have:
1 1
long = ( --------------- + 1 ) sin(phi) + --------------- phi
2 2
( sin(lat) ) ( sin(lat) )
and this function has no analytic inverse, so we must deal with it as we had dealt with latitude as a function of y for the Mollweide and Eckert IV projections, determining phi by an iterative process.
On the Equator, the scale of 1 + (x ^ 2)/2 gives, when integrated:
long = x + (x ^ 3)/6
which is invertible, being a cubic equation.