The general method of constructing de veloped projections has been briefly described under the title MAP in this Encyclopedia; but, as all modern maps representing data ob tained from precise trigonometrical surveys are based upon some form of developed projection, the mathematical theory and the practical meth ods of construction of such will receive more extended treatment herein.
The various projections form three definite groups—(1) those possessing straight meridians and parallels, (2) those of mixed systems of straight and curved meridians and parallels, and (3) those in which both meridians and parallels appear as curved lines.
Of the first group, the one most extensively used is Mercator's projection. It was devised to satisfy the following condition: That the loxodromic curve or the course of a ship on the surface of the sea, under a constant bearing or intersecting the successive meridians at the same angle, shall appear on the projection as a straight line having the same angle of bearing with respect to the meridians intersected as that of the loxodromic curve.
The formula used in computing a table of emeridional parts° or the °increased latitudes° for determining the distances of the various parallels, of projecting latitudes from the equator, is obtained as follows: (Modified ex tract from Projection Tables published by the United States Hydrographic Office).
Referring to Fig. 1, let 1 c be an element of the loxodromic curve between two consecutive meridians m e m' e'; and let L C represent the corresponding element on the projection between the corresponding meridians M E, M' E'. Let 1 p and L P be taken parallel to the corresponding equatorial elements e e' and E E' between the same meridians.
Then, the condition that the angles of ing 1 c p and L C P shall be equal requires that CP LP p or since L P --= E e e the element on the terrestrial equator, it is necessary that C P e e c Hence, putting d s r the meridional element c p of the terrestrial spheroid, d m for the meridional element C P of the projection, a for the equatorial radius of the earth, and r for the radius of the parallel represented by the element I p; then, on account of the proportion ality of the elemental arcs e e; I p, to their respective radii a, r, we have the fundamental equation d a d s r which expresses the law of the Mercator pro jection.
Then, if L be the latitude of the terrestrial parallel under consideration, R the radius of curvature of the terrestrial meridian at its point of intersection with the parallel, c the compression of the earth, and e its meridional eccentricity; we have the folowing expression for the properties of r, R, and e of the terres trial spheriod considered as an ellipsoid of revolution: a — cos L r — (I —es sin' L) a (1—e') R= ( 1— e' sirr' L) e= V 2c — c' .
Now, since the radius of curvature varies inversely as the angle between consecutive normals, the element of the terrestrial meridian at its intersection with any parallel of latitude is equal to the product of the radius of curva ture and the element of latitude at that point, and ds = R. dL, which being substituted in the fundamental equation gives us the expression a R. dL dm =--- r for the element of the projected meridian, or by substituting the preceding values of r and R, we have a (1— dL dm— (1--e L) cos L'which when integrated between the proper limits give required length of any finite portion of the projected or chart meridian cor responding to the meridional arc on the ter restrial spheroid within the same limits.
For this integration, multiplying e' in the numerator by sin L cos'• L, we have a. dL cos L. dL dm — cos L 1— sin' again, multiplying the numerator and denomina tor of the first term by cos L, substituting 1—sin' L for cos' L, and resolving both terms into partial fractions, we have the expression =__ a I cos L cos L dL — ae dm 2 kl -1- sin L 1— sin L 2 e cos Le cos L e sin 1— e sin L/ whence, by performing the integration or the limits of 0 and L, we find for the length of that part of the meridian of the projection in cluded between the equator and the parallel of latitude L, the expression =__ a (1 log 1+ sin L 1 1+ e sin L) M 1— sin L 2 s log 1—e sin L in which the logarithms belongs to the common system, of which M is the modulus.