202 . '-‘ . 7 . 5111?‘ u‘? 1-,’, + Slll u cos 1: ‘(ii — _r/= u. — sin 11, - ¢ :1 dd 51'’ = i <rIl()B (- This method of development is due to Sir George Airy, whose original paper—the irwestig-.1tion is different in form from the above—will be found in the Philosophical Jfaguziue for December 1861. The solution of the differ- ential equation leads to this result— it at ‘It P52 cot 310g. sec 5 + Ctan 2- , n..1~a_-, l C=2 cot‘-jg low see c‘ .' The limiting radius of the map is P»: "Ctan1,B. In this system, called by the Astronomer—Royal the “ 1’rojectz'on by balance of errors,” the total misrepresentation is an absolute minimum. Returning to the general case where p is any function of u, let us consider the local misrepresentation of direc- tion. Take any indefinitely small line, length =27, making an angle a with the meridian in co—latitude 21.. Its projec- tions on a meridian and parallel are {cos a, {sin a, which in the map are represented by io- cosa, io-'sin a. If then a.’ be the angle in the map corresponding to a, I 0’ tan a'=—— tan a. 0’ Put and the error a’ — a of representation = 6, then (2 — 1) tan a tan e = __ 1 + 2 tan- :1. Put E= cot‘-’ {, then 5 is a maximum when a=§, and the corresponding value of e is e =? — QC . For simplicity of explanation we have supposed this method of development so applied as to have the pole in the centre. There is, however, no necessity for this, and any point on the surface of the sphere may be taken as the centre. All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meri- dians and parallels within the space which is to be repre- sentedin aplaue. Then the azimuth is represented unaltered, and any spherical distance it is represented by p. Thus we get all the points of intersection transferred to the repre- sentation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. The exaggeration in such systems, it is important to re- member, whether of linear scale, area, or angle, is the same for a given distance from the centre, whatever be the azimuth; that is, the exaggeration is a function of the dis- tance from the centre only. We shall now examine and exemplify some of the most important systems of projection and development, commen- cing with Perspective Projections. In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane perpendicular to the line joining the centre of the sphere and the point of vision. If V be the point of vision, P any point on the spherical surface, then 1), the point in which the straight line VP intersects the plane of the representation, is the projection of P. In the orthographic projection, the point of vision is at an infinite distance and the rays consequently parallel ; in this GEOGRI—’lIY ' case be ellipses described on was as [.I.vr1n-;.A'r1eAr.. case the plane of the drawing may be supposed to pass through the centre of the sphere. Let the circle (fig. 8) represent the plane of the equator on which we propose to make an orthographic representation of meridians and parallels. The centre of this circle is clearly the projection of the pole, and the parallels are projected into circles having the pole for a common centre. The diameters aa’, bl)’ being at right angles, let the semicircle Lab’ be divided into the required number of equal parts ; the diameters drawn throngh these points are the projec- tions of meridians. The distances of c, of J, and of e from the diameter art.’ are the radii of the successive circles representing the parallels. It is clear that, when the points of division are very close, the parallels will be very much crowded towards the outside of the map; so much so, that this projection is not much used. For an orthographic projection of the globe on a meridian plane, let qnrs (fig. 9) be the meri- n L dian, as the axis of rotation, then qr °_d X e is the'projection of the equator. The parallels will be represented by straight lines passing through the 9 " points of equal division; these lines T are, like the equator, perpendicular I 1/ to us. The meridians will in this % a common major axis, the distances 173- 9- of c, of (I, and of e from us being the minor semiaxes. Let us next construct an orthographic projection of the sphere on the horizon of any place. Set off the angle «op (fig. 10) from the radius oa, equal to the latitude. Drop the perpendicular pl’ on oa, then 1’ is the projection of the pole. 011 (L0 produced take ob=7;P, then ob is the minor semi- axis of the ellipse representing the equator,_its major axis being gr at right angles to ‘((0. The points in which the meridians meet this ellip- tic equator are determined by lines drawn parallel to (cob through the points of equal subdivision cclef}//L. Take two points, as cl and g, which are 90° apart, and let 27: be their projections on the equator; then i is the pole of the meridian which passes through Ir. This meridian is of course an ellipse, a11d is described with refer- ence to i exactly as the equator was described with refer- ence to P. Produce 270 to /, and make lo equal to half the shortest chord that — can be drawn through i ; then [0 is the semi- axis of the elliptic meridian, and the major axis is the dia- meter perpendicular to iol. For the parallels: let it be required to describe the parallel whose co-latitude is u ; take pm =1)». = u, and let m'n' be the projections of m and n on 0Pa ; then m'n' is the minor axis of the ellipse representing the parallel. Its centre is of course midway between m’ and n’, and the greater axis is equal to ma. Thus the construction is ob- vious. Wheu pm is less than pa, the whole of the ellipse Fig. 10.
FIG. 11. -01-tlxograpliic Projection.