(a) The errors of scale of the extreme parallels may be made equal and may be equated to the error of scale of the parallel of maximum error (which is near the mean parallel). (b) Or the errors of scale of the extreme parallels may be equated to that of the mean parallel. This is not so good a projection as (a). (c) Or the absolute errors of the extreme and mean parallels may be equated. (d) Or in the last the parallel of maximum error may be considered instead of the mean parallel. (e) Or the mean length of all the parallels may be made correct. This is equivalent to making the total area between the extreme parallels correct, and must be combined with another condition, for example, that the errors of scale on the extreme parallels shall be equal. We will now discuss (a) above, viz. a conical projection with rectified meridians and two standard parallels, the scale errors of the extreme parallels and parallel of maximum error being equated. Since the scale errors of the extreme parallels are to be equal, h(z + c) h(z + c´) c´ sin c - c sin c´ -------- - 1 = --------- - 1, whence z = ------------------- (i.) sin c sin c´ sin c´ - sin c The error of scale along any parallel (near the centre), of which the co-latitude is b is 1 - {h(z + b)/sin b}. (ii.) This is a maximum when (tan b) - b = z, whence b is found. Also h(z + b) h(z + c) 1 - -------- = -------- - 1 whence h is found. (iii.) sin b sin c For the errorless parallels of co-latitudes [gamma] and [gamma]´ we have h = (z + [gamma])/sin [gamma] = (z + [gamma]´)/sin [gamma]´. If this is applied to the case of a map of South Africa between the limits 15° S. and 35° S. (see fig. 16) it will be found that the parallel of maximum error is 25° 20´; the errorless parallels, to the nearest degree, are those of 18° and 32°. The greatest scale error in this case is about 0.7%. In the above account the earth has been treated as a sphere. Of course its real shape is approximately a spheroid of revolution, and the values of the axes most commonly employed are those of Clarke or of Bessel. For the spheroid, formulae arrived at by the same principles but more cumbrous in shape must be used. But it will usually be sufficient for the selection of the errorless parallels to use the simple spherical formulae given above; then, having made the selection of these parallels, the true spheroidal lengths along the meridians between them can be taken out of the ordinary tables (such as those published by the Ordnance Survey or by the U.S. Coast and Geodetic Survey). Thus, if a1, a2, are the lengths of 1° of the errorless parallels (taken from the tables), d the true rectified length of the meridian arc between them (taken from the tables), h = {(a2 - a1)/d}180/[pi],