G-E]()]D E S'Y cosines of l}.[ are therefore proportional to l’ -I : m’ — m : n’ — 2:. If the angle made by BC with AB be indefinitely small, the direction cosiiies of BM are as Bl : Sm: Sn. N ow if AB, BC be two contiguous elements of a geodesic, then ]3.[ must be a normal to the surface, and since 3!, 8m, Sn are in this case represented by 3d—x, Sill/-, 8&3, we have (ls (Is (ls (13.7; d‘~’_i/ (lg: (E _ _ (79 17:7 _ 7:7 " E’ (T: @ dz which, however, are equivalent to only one equation. In the case of the splieroid this equation becomes (l".t' fl? 1/ _ ytT"i _ :L(lsT-’ -0 ’ which integrated gives 3/cI.v—x¢l_r/= Ctls. This again may be put in the.forn1 r sin a.= C, where a. is the azimuth of the geodesic at any point—the angle between its direction and that of the iiieridian—and r the distance of the point from the axis of revolution. From this it may be shown that the aziiniith at A of the geodesic joining AB is not the same as the astronomical azimuth at A of B or that determined by the vertical plane Aal3. Generally speaking, the geodesic lies between the two plane section curves joining A and B which are formed by the two vertical planes, supposing these points not far apart. If, however, A and B are nearly in the same latitude, the geodesic may cross (between A and B) that plane curve which lies nearest the adjacent pole of the spheroid. The condition of crossing is this. Suppose that for a moment we drop the consideration of the earth’s iion—spliericity, and draw a perpendicular from the pole C on AB, meeting it in S between A and B. Then A being that point which is nearest the pole, the geodesic will cross the plane curve if AS be between -_}AB and §»Al3. If AS lie between this last value and §Al3, the geodesic will lie wholly to the north of both plane curves, that is, supposing both points to be in the northern hemisphere. The circuinstaiice tliit the angles of the geodesic triangle do not coincide with the true angles as observed renders it inconvenient to regard the geodesic lines as sides of the triangle. A more convenient curve to regard as the side of the spheroidal triangle is this: let L be a point on the curve surface between A and B, A the point in which the normal at L intersects the axis of revolution, then if L be subject to the condition that the planes ALA, BLA coincide, it traces out a curve which touches at A and B the two plane curves before specified. Joining A, B, C by three such lines, the angles of the triangle so formed coincide with the true angles. Let the azimuths (at the middle point, say) of the sides BC, CA, All of a spheroidal triangle be a , B , -y , these being measured from 0° to 360" continuouslv, and the angles of the trianrrle lettei'ed in the same cyclical direction, and let a, b, c be the lengtlijs of the sides. Let there be a sphere of radius 7', such that 7' is a mean proportional between the principal radii of curvature at the mean latitude 4) of the spheroidal triangle, and on this sphere a triangle having sides equal i'espeetively to a, b, c. If A’, B’, C’ be the angles of the spheroidal triangle, A, B, C those of the spherical triangle, then c" _.. .. . . . _2 cos -<p(b- sin ‘.33 — c3 sin 2-yl , w—A='" 12) D’—- B = —1_j):_J cos ‘-’<;b(c‘3 sin 53-): — a'-' sin Ea) , -2 C’ — C=—1£)—_, cos ‘3<p(a‘-’ sin 2:; — IF sin 23) . 5-.)-4 I By adding these together, it appears that, to the order of ternis ieie retained, the sum of the angles of the spheroidal triangle is equal_ to the sum of the angles of the spherical triangle. The Sl’l‘°1'“33l CXCCSS Of a spheroidal triangle is therefore obtained by . . . 1 multiplying its area by W , Gauss's measure of curvature. 169 _Further, let Al, ril, C, be the angles of a plane triangle having still the same sides a, b, c, then it may be shown by spherical trigo. iioinetry that, 1' being the radius of the sphere as before, A_A = A!" 1+(_t‘3+7b2+7c'-’ ‘ 3r- 12or2 ’ A 7a2+b‘3+7c‘3 B—D1=§rT‘(1+- 1207'” )' . . A 7a2+7b’+c" C‘ (‘=31-'-(1 J’ 1-;’02-2 W) It is but seldom that the terms of the fourth order are required. Oiiiittiiig them, we have Legendre’s theorem, viz., “ If from each of the angles of a spherical triangle, the sides of which are small in comparison with the radius, one—tliird of the spherical excess be deducted, the siiies of the angles thus diminished will be proportional to the length of the opposite sides, so that the triangle may be computed as a plane triangle.” By this means the spherical triangles which present themselves in geodesy are computed with very nearly the same ease as plane triangles. And from the expressions given above for the spheroidal angles A’, D’, C’ it may be proved that no error of any consequence can arise from treating a spheroidal triangle as a spherical, the radius of the sphere being as stated above. When the angles of a triangulation have been adjusted by the method of least squares, the next process is to calcu- late the latitudes and loiigitudes of all the stations starting from one given point. The calculated latitudes, longitudes, and azimuths, which are designated geodetic latitudes, longitudes, and aziinuths, are not to be confounded with the observed latitudes, longitudes, and aziniutlis, for these last are subject to somewhat large errors. Supposing the latitudes of a number of stations in the triangulation to be observed, practically the mean of these determines the posi- tion in latitude of the network, taken as a whole. So the orientation or general azimuth of the whole is inferred from all the azimuth observations. The triangulation is then supposed to be projected on a splieroid of given elements, representing as nearly as one knows the real figure of the earth. Then, taking the latitude of one point and the direc- tion of the meridian there as giveii—obtained, namely, from the astronoinical observations there—one can compute the latitudes of all the other points with any degree of precision that may be considered desirable. It is necessary to employ for this purpose formulae which will give results true even for the longest distances to the second place of decimals of seconds, otherwise there will arise an aecuinulation of errors from imperfectcalculation which should always be avoided. For very long distances, eight places of decimals should be employed in logarithmic calculations; if seven places only are available very great care will be required to keep the last place true. N ow let <35, <35’ be the latitudes of two stations A and B ; a, or.’ their mutual azimuths counted from north by east continuously from 0° to 360° ; (.0 their difference of longitude measured from west to east; and s the distance AB. First compute a latitude <35, by means of the formula 3 . . qS1=qS—:-—cos a, where p is the radius of curvature of the P iiieridiaii at the latitude «,1; ; this will require but four places of logarithms. Then, in the first two of the following, five places are sufficient- e=_S‘ sin a cos a, 3'3 . . -q=— sin 3a tan zp, , "pit zp -<p=P3 cos (a.-—§e)-17 , =s_ sin (a-—§,s) 7L cos(<,‘l/+511)’ O a'—a=wSln (¢'+§'r_i)-£+130. _ 1. — 22
(A)