The radius of the sphere is assumed to be unity, and z and γ are expressed in circular measure. Hence h = sin γ/(z + γ) = sin γ′ (z + γ′); from this h and z are easily found.
In the above description it has been assumed that the two errorless parallels have been selected. But it is usually desirable to impose some condition which itself will fix the errorless parallels. There are many conditions, any one of which may be imposed. In fig. 15 let Cm and C′m′ represent the extreme parallels of the map, and let the co-latitudes of these parallels be c and c′, then any one of the following conditions may be fulfilled:—
(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) | − 1 = | h (z + c′) | − 1, whence z = | c′ sin c − c sin c′ | . |
| sin c | sin c′ | sin c′ − sin c |