206 but any extent in the direction of a parallel. Selecting ' the mean parallel, or that which most nearly divides the ' area to be represented, we have to consider the cone which touches the sphere along that parallel. In fig. 20, which is an orthographic projection of the sphere. on a meridian plane, let Pp be the parallel of contact with the cone. ON being the axis of revolution, the tangents at P and 72 will intersect ON pro- duced in V. Let Q7 be a parallel to the north of Pp, Br another parallel the same dis- tance to the south, that is, PQ=PR. Take on the tan— _ 0 gent PV two points H, K such P13‘ *0‘ that PH =PK, each being made equal to the arc PQ. It is clear, then, that the surface generated by Hli is very nearly coincident with the surface generated by RQ when the figure rotates round ON through any angle, great or small. The approximation of the surfaces will, however, be very close only if QR is very small. Suppose, now, that the paths of H and K, as described in the revolution round ON, are actually marked on the surface of the cone, as well as the line of contact with the sphere. And further, mark the surface _,~‘j._ of the cone by the intersections with it " " of the meridian planes through OV at the required equal intervals. Then let the cone be cut along a generating line and opened out into a plane, and we shall have a representation as in fig. 21 of the spherical surface con- tained between the latitudes of Q and Fig. 21. R. The parallels here are represented by concentric circles, the meridians by lines drawn through the common centre of the circles at equal angular intervals. Taking the radius of the sphere as unity, and gb being the latitude of P, we see that VP=cot <1», and if L0 be the difference of longitude between two meridians, the corresponding length of the arc Pp is wcos <1». The angle between these meridians them- selves is co sin #3. Suppose, now, we require to construct a map on this prin- ciple for a tract of country extending from latitude ¢> — m to ¢+m, and covering a breadth of longitude of 2n, m and n being expressed in degrees. In fig. 21 let HKZ-/a be the quadrilateral formed by the extreme lines, so that HK =k/t°= 9m; then the angle HVIL is 212. sin (1: expressed in degrees. N ow, takingthe length of a degree as the unit, VP = 57'296 cot gb, and VH = 57296 cot gb — m. It may be convenient in the first instance to calculate the chords H/2, Kk, and thus construct the rectilinear quadrilateral HKl'/z. The lengths of these chords are Hh=2(57'296 cot ¢;b-on) sin (n sin ¢>) , Klc=2(57'296 cot ¢+m) sin (a sin :1») , and the distance between them is 2m cos (72 sin (1)). The inclined sides of this trapezoid will then meet in a point at V, whose distance from P and 7; must corre- spond with the calculated length of VP. N ow with this centre V describe the circular arcs representing the parallels through H, K, P. Also if the parallels are to be drawn at every degree of latitude, divide HK into 2122 equal parts, and through each point of division describe a circular are from the centre V. Then divide Pp into 27¢ equal parts, and draw the meridian lines through each of these points of division and the centre V. If the centre V be inconvenierrtly far off, it may be necessary to construct the centre parallel by points, that is, by calculating the coordinates of the various points of V N H {F-‘ [I O I . - - ' - n (1 E0 (i It A l’ ll Y division. For this purpose, draw through the intersection [.r,-.'rrrr:_r_-vr1cAL. of the centre meridian and centre parallel a line perpendi- cular to the meridian and therefore tonclring the parallel. Let the coordinate .-2; be measured from the centre along this line, and 3/ perpendicular to it. Then the coordinates of a point whose longitrrde measured from the centre meridian is co are
- c: cot «t sin (0: sin ¢),
2 :2cot ¢ sin’-’§(w sin ¢):a' tan {Aw sin ¢), the radius of the sphere being the unit; if a degree he the unit, these nmst be multiplied by 57296. The great defect of this projection is the exaggeration of the lengths of parallels towards either the northern or southern limits of the map. Various have been the devices to remedy this defect, and amongst these the following is a system very much adopted. llav- ing subdivided the central meridian and / drawn through the points of division the parallels precisely as described above, then the true lengths of degrees are Fir} 33- set off along each parallel 5 the meridians, which in this case become curved lines, are drawn through the corre- sponding points of the parallels (fig. 22). This system is that which was adopted in 1803 by the “ Dépot de la Guerre ” for the map of France, and is there known by the title “Prq}'er_'tion do Iiomze." It is that on which the Ordnance Survey map of Scotland on the scale of one inch to a mile is constructed, and it is frequently met with in ordinary atlases. It is ill-adapted for countries having great extent in longitude, as the intersections of the meridians and parallels become very oblique—as will be seen on examining the map of Asia in most atlases. If qbu be taken as the latitude of the centre parallel, and co-ordinates be measured from the intersection of this parallel with the central meridian, as in the case of the conical projection, then, if p be the radius of the parallel of latitrrde gt), we have p= cot gbo + qbo — gb. Also, if S be a point on this parallel whose co-ordinates are .1, 3/, so that VS=p, and 9 be the angleVS makes with the central meridian, then p0 = or cos gb; and .':::p sin 0, 3/:cot qbo — p cos 0. Now, if we form the differential coefiicients of ac and 3/ with respect to gb and co, the latitrrdc and longitrrde of S, we get on’-n — mu’: cos ¢, 'mn+m’n’: ‘—" cos¢>(cos ¢> — p sin ¢) ; P the first of which equations proves that the areas ar'e truly represented. Moreover, if 90° =!= 1,11 be the angles of intersec- tions of meridians and parallels, tan rp_0 — co sin 96, which indeed might have been more easily obtained. In the case of Asia, the middle latitude §b°= -10°, and the ex- treme northern latitude is 70°. Also the map extends 90 of longitude from the central meridian; hence, at the north- west and north-east corners of the map the angles of inter- section of meridians and parallels are 90° =5 33°'5-1’. Ilut for comparatively small tracts of country, as France or Scot- land, this projection is very'suitable. Another modification of the conical projection consists in taking, not a tangent cone, brrt a cone which, having its vertex in the axis of revolution produced, intersects the sphere in two parallels,—these parallels being approximately midway between the centre parallel of the country and the extreme parallels. By this means part of the error is thrown on the centre parallel which is no longer represented by its true length, but is made too small, while the parallels forming the intersections of the cone are truly re- presented in length.
The exact position of these particular parallels may be
