Substituting for 1+ sin L 1— sin L its trigonometrical equivalent, and for log 1 e sin L 1— e sin L its algebraic development, we have the final expression tan (45° — a sin L I sins L I el sing L.) which is the formula required for computing the meridional parts for the terrestrial spheroid.
This formula consists of two parts— the first term being the ordinary formula for computing meridional parts with the earth regarded as a sphere, while the second term consists of a series of negative terms, and represents the correction which must be applied for the meridional eccentricity of the earth under the hypothesis that it is an ellipsoid of revolution.
In using the formula to compute the ac companying table, the following numerical values with their respective logarithms have been employed for the several constants.
The equatorial radius is expressed in minutes of arc, therefore, the unit of measure of the meridional part m becomes a minute of the equator or a geographical mile.
10,800' a = =3437'.74677 log 3.5362739.
The reciprocal of the modulus of the com mon logarithms, 1 — =2.3025851 log 0.3622157.
Bessel's determination of the compression, 1 c.— =0.003342773 log 7.5241069.
100.1528 The meridional eccentricity of the earth, e.= y 2c-c' = 0.0816968 log 8.9122052.
The values of the several coefficients with their logarithms deduced are as follows: a — = 7915%7055 log 3.8984896 M ae'= 22'.9448 log 1.3606843 ae' = 0%05104731 log 8.7079734 0'.000204425 log 6.3105351 On account of the rapid convergence of the series of coefficients involving the eccentricity, two terms are generally sufficient for practical use, and the formula becomes L m =7915'.7055 log tan 45° + —22'.9448 2 sin L 0'.05104731 sin' L.
On account of the space limitations the table gives the meridional parts for even degrees of latitude only; the values for intermediate minutes or seconds of arc may be computed by the formula.
If the chart for which the projection is made includes the equator, the values given may be measured off for the successive degrees of latitude directly from the equator; but, if the. equator is not included in the map, the parallels of latitude to be projected should be succes sively measured from a principal parallel, preferably the lowest parallel drawn upon the map, and the distance of any parallel from the principal parallel will be the difference of the values given for the two in the table.
These values, given in minutes of arc, may be converted into their equivalents in inches, yards, meters, etc., and laid off on the projection by means of properly divided corresponding scales, proportionately to the scale adopted for the map, or the values given may be laid off without previous numerical conversion, by means of a diagonal scale constructed on the map.
For example—suppose a Mercator projec tion is required to embrace the coasts of Ice land, on a scale of 1-inch..= 150 statute miles. This island lies between latitudes 63° and north, and between longitudes 13° and west from Greenwich. The projection will in clude four degrees of latitude and 12 degrees of longitude. The central meridian will cor respond to longitude 19° W., and the lowest parallel will correspond to latitude 65° N. See Fig. 2.
Draw in the centre of the sheet a vertical statute miles; 660.72 ÷ 150=4.40 inches or the distance of the Parallel of 67° from the parallel of 63° on the map, on a scale of 1-inch straight line A B, for the central meridian, and near the bottom of the map construct very fully a horizontal line C D, at right angles to A B, for the parallel of 63° N., and assume it as the principal parallel of the map. From the table obtain the value 4884'.46 for lat. 63° and 5452'.84 for lat. 67°, the differ ence of which or 5452'. 84 — 4884'.46 = 568'.38, the value of the meri dional arc included be tween the lowest and the highest parallels of latitude of the map, for which one minute of arc of the equator 1.012 statute miles is taken a, the unit of measurement. Convert ing this value to its equivalent in statute miles and reducing it tt the scale of the map, gives 568'.38 60'= 9°.47; 9°.47 X 69.77 statute miles (equivalent ti a degree of longitude on the equator) = 150 statute miles. It is obvious that this 'measurement may be laid off on the map by means of any scale di vided to the decimals of an inch, and the dis tance of each of the other parallels from the principal may be ob tained by the same method of conversion, and laid off in a similar manner; butt, having ob tained the value 4.40 inches equal to the ver tical extent .)f the map, we are to de termine a nstant mul tiplier by the use of which the tabular values may be laid off directly by means of a diagonal scale (q.v.) of one inch without pre vious numerical conversion. As 1' of arc of the equator is taken as the unit of the meridional arc, 1' of arc of latitude will measure 4.40 indies 568'.38=0.0077 inches, which corre sponds to the scale of the map and by which all the values obtained from the table must be multiplied if they are to be laid off on the projection by means of a diagonal scale of one inch.