(i.)
we can dispose of C and h so that any two selected parallels shall be their true lengths; let their co-latitudes be z1 and z2, then
2h (C − cos z1) = sin2 z1
(v.)
2h (C − cos z2) = sin2 z2
(vi.)
from which C and h are easily found, and the radii are obtained from (i.) above. This is H. C. Albers’ conical equal-area projection with two standard parallels. The pole is not the centre of the parallels.
Projection by Rectangular Spheroidal Co-ordinates.
If in the simple conical projection the selected parallel is the equator, this and the other parallels become parallel straight lines and the meridians are straight lines spaced at equatorial distances, cutting the parallels at right angles; the parallels are their true distances apart. This projection is the simple cylindrical. If now we imagine the touching cylinder turned through a right-angle In such a way as to touch the sphere along any meridian, a projection is obtained exactly similar to the last, except that in this case we represent, not parallels and meridians, but small circles parallel to the given meridian and great circles at right angles to it. It is clear that the projection is a special case of conical projection. The position of any point on the earth’s surface is thus referred, on this projection, to a selected meridian as one axis, and any great circle at right angles to it as the other. Or, in other words, any point is fixed by the length of the perpendicular from it on to the fixed meridian and the distance of the foot of the perpendicular from some fixed point on the meridian, these spherical or spheroidal co-ordinates being plotted as plane rectangular co-ordinates.
The perpendicular is really a plane section of the surface through the given point at right angles to the chosen meridian, and may be briefly called a great circle. Such a great circle clearly diverges from the parallel; the exact difference in latitude and longitude between the point and the foot of the perpendicular can be at once obtained by ordinary geodetic formulae, putting the azimuth = 90°. Approximately the difference of latitude in seconds is x2 tan φ cosec 1″ / 2ρν where x is the length of the perpendicular, ρ that of the radius of curvature to the meridian, ν that of the normal terminated by the minor axis, φ the latitude of the foot of the perpendicular. The difference of longitude in seconds is approximately x sec ρ cosec 1″ / ν. The resulting error consists principally of an exaggeration of scale north and south and is approximately equal to sec x (expressing x in arc); it is practically independent of the extent in latitude.