The distances of the successive parallels from the parallel of thus obtained are as follows: 0.0077 X (501&76 — 4884.46) = 1.03 in. (64°) 0.0077 X (5157.98 — 4884.46) = 2.11 in. (61 0.0077 X (5302.51 — 4884.46) = 322 in. 0.0077 X (5452.84 — 4884.46) = 4.40 in. These distances may now be laid off upon the central meridian from its intersection with the principal parallel in the order 1, 2, 3, 4 shown on Fig. 2, and through the points thus obtained straight lines may be drawn parallel to the principal parallel, and corresponding to the terrestrial latitudes of 65°, and 67°.
On this map, a degree of longitude will measure 0.0077 inches X 60=0A6 inches and the distances of the successive full degrees of longitude east and west of the central meridian will be 0.46, 0.92, 1.39, 1.83, 2.31, 2.77 inches. Lay off these values on the lowest, middle and highest parallels of latitude, east and west from the central meridian, in the order a, b, c, d, e, f, and through the points thus obtained draw straight lines parallel to the central meridian and corresponding to the terrestrial longitudes of 13° 14' 15°, 16°, 19°, 20 , 21°, 22°, 23°, and 25°, west from Greenwich. Another important projection in which both meridians and parallels are represerned by straight lines is the cyclindrical equal area projection, formed by projecting the surface of the earth from its axis on to a cylinder tangent at the equator and unrolling. While showing extreme distortion in regions remote from the equator, this map has the valuable property of representing areas truly.
The second group of projections, those hav ing mixed systems of straight and curved merid ians and parallels, including various kinds of ((equal and projections, of which the a simple conic" is the most valuable for general purposes. In this projection it is assumed that a cone, the apex of which lies in the axis (produced) of the sphere, is tangent to the surface of the sphere along a parallel of lati tude. When the surface of the cone is de veloped on a plane, the parallel of tangency be comes an arc of a circle, having for its radius the slant side of the cone which is equal to the cotangent of the latitude. A part of this de veloped arc of a circle, of sufficient length to include the desired number of degrees of the proposed map, is drawn through the middle point of the central meridian and forms the middle parallel of the map. The central merid ian north and south of this line is divided into degrees of latitude laid off according to scale proportional to their true lengths on the sphere, and parallels of latitude concentric to the mid dle parallel are drawn through the several points thus obtained. In a similar manner, the middle parallel is divided into degrees of longi tude east and west of the central meridian and straight lines representing meridians are drawn through those points to the centre from which the concentric parallels were swept. In the sra
tem of co-ordinates thus established, the parallels and meridians intersect each other at right angles, precisely the same as on the sphere. The lengths of the degrees on the central meridian and on the middle parallel are proportionately the same as those on the sphere, and the lengths of the degrees on all the other meridians and parallels are but slightly different from what they would be if laid off their true length proportioned to those of the sphere.
In general, a conical projection is one which transforms the meridians into a pencil of straight lines, the angle between two of which is in a fixed ratio to the angle between the two meridians which they represent. The parallels will then form a family of concentric circles, and the particular form of the projection will be determined by the relation which the radius of such a circle will bear to the polar distance of the corresponding parallel. Let the radius of the circle be r. the polar distance o, the angle between a meridian and a fixed n'eridian of reference on the sphere ben and on the map 4, Let our unit ot distance be the ranius of the earth. It is clear that the method of mapping will be completely determined by the equations r =f (P) .=-- c O.
The scale of representation of meridian dis tances on the map at (-pi, 0s) will be of (pi) dP The scale of representation of distances along the parallels will be OLCI-'11 If our scale of sin ' representation is to be the same in all tions, r = k (tan c, where k is an arbitrary 2 constant. This gives what is known as die orthomorphic conical projection. If the scale representation at a point p remains fixed, while c decreases toward 0, the map will approach a mercator projection. If c —1, the angles be tween meridians on the map will equal those on the earth, and we shall have the stereographic projection. If no matter what relation the meridian and parallel scales bear to one another, our projection is said to be zenithal, as are similar projections, where other families of concurrent great circles replace the meridians. If the meridian scale of a zenithal projection is fixed we have what is known as the equi distant projection; if the projection is coni cal but not zenithal it is said to have rectified meridians. If the meridian and parallel scales of a conical projection are reciprocals, areas are preserved proportionally. If the projection is also zenithal it is sometimes called Lorgna's projection. A form of zenithal pro jection in which the deformation is distributed and minimized is due to Sir George Airy.