Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/627

EAR T H 605 tance it is 637 feet, and at three quarters it is 473 feet. The azimuth of the geodesic at Cadiz differs 20" from that of the vertical plane, which is the astronomical azimuth. The azimuth of a geodesic line cannot be observed, so that the line does not enter of necessity into practical geodesy, although many formulas connected with its use are of great simplicity and elegance. The geodesic line has always held a more important place in the science of geodesy among the mathematicians of France, Germany, and Kussia than has been assigned to it in the operations of the English and Indian triangulations. Although the observed angles of a triangulation are not geodesic angles, yet in the calculation of the distance and reciprocal bearings of two points which are far apart, and are connected by a long chain of triangles, we may fall upon the geodesic line in this manner : If A, Z be the points, then to start the calculation from A, we obtain by some preliminary calculation the approximate azimuth of Z, or the angle made by the direction of Z with the side AB or AC of the first triangle. Let P l be the point where this line inter sects BC; then, to find P 2 , where the line cuts the next triangle side CD, we make the angle BP X P 2 such that BP 1 P 2 + BP 1 A = 180. This fixes P 2 , and P 3 is fixed by a repetition of the same process ; soforP 4 , P 5 . . . . Now it is clear that the points P lf P 2 ,P 3 so com puted are those which would be actually fixed by an observer with a theodolite, proceeding in the following manner. Having set the instrument up at A, and turned the telescope in the direction of the computed bearing, an assistant places a mark Pj on the line BC, adjusting it till bisected by the cross-hairs of the telescope at A. The theodolite is then placed over P l , and the telescope turned to A ; the horizontal circle is then moved through 180. The assistant then places a mark P 2 on the line CD, so as to be bisected by the telescope, which is then moved to P 2 , and in the same manner P 3 is fixed. Now it is clear that the series of points P! , P 2 , P 3 approaches to the geodesic line, for the plane of any two consecutive elements P n _i P n , P n P n+ i contains the normal at P n . From the formulae which we have given above, expressing the mutual relations of two points P, Q on a spheroid, we may obtain the following solution of the problem : Given the latitude <j> of P, with the azimuth a and distance s of Q, to determine the latitude and longitude of Q and the back azimuth a . Let - 4(1 - e 2 ) e-6 3 cos 2 <t> sin 2 cos 2 <p cos 2 a ; ~6(l-e 2 ) f are always veryminute quantities even for the longest distances ; then, putting = 90- ^, sin " tan a +C+w. t 2 tan J here p is the radius of curvature of the meridian for the mean latitude J(<f> + < ) These formula are approximate only, but they are sufficiently precise even for very long distances. Meridian Arcs. The length of the arc of meridian between the latitudes <t> t and <p t instead of using the excentricity, put the ratio of the axes 1 -n : l+n , then = / J 0i This, after integration, gives where a o-^?-^i aj = sin (< 2 - c^j) cos (< 05= sin 2((kj- <J>i) cos 03= sin 3(^2 -<j>i) cos The part of s which depends on n 3 is very small ; in fact, if we calculate it for the longest arc measured, the Kussian arc, it amounts to only an inch and a half, therefore we omit this term, and put for T the value Now, if we suppose the observed latitudes to be affected with errors, and that the true latitudes are <(> l + x l , 4> z + a: 2 ; and if further we suppose that n l + dn is the true value of a-b : a + b, and that n 1 itself is merely a very approximate numerical value, we get, on making these substitutions and neglecting the influence of the cor rections x on ike position of the arc in latitude, i.e., on here da =x^- x ; and as b is only known approximately, put b 1 = b(l+u); then we get, after dividing through by the coefficient of da Q , which is = 1 + % - 3% cos (</> a - <Pi) cos (<f> 4 - fa), an equation of the form x i =x 1 + h+fu + gv, where for convenience we put v for dn. Now in every measured arc there are not only the extreme stations determined in latitude, but also a number of intermediate stations, so that if there be i + 1 stations there will be i equations In combining a number of different arcs of meridian, with the view of determining the figure of the earth, each arc will supply a number of equations in u and v and the cor rections to its observed latitudes. Then, according to the method of least squares, those values of u and v are the most probable which render the sum of the squares of all the errors x a minimum. The corrections x which are here applied arise not from errors of observation only. The mere uncertainty of a latitude, as determined with modern instruments, does not exceed a very small fraction of a second as far as errors of observation go, but no accuracy in observing will remove the error that may arise from local attraction. This, as we have seen, may amount to some seconds, so that the corrections x to the observed latitudes are attributable to local attraction. Archdeacon Pratt, in his treatise on the figure of the earth, objects to this mode of applying least squares first used by Bessel ; but certainly Bessel was right, and the objection is groundless. Comparisons of Standards. In determining the figure of the earth from the arcs of meridian measured in different countries, one source of uncertainty was, until the last few years, the want of com parisons between the standards of length in which the arcs were expressed. This has been removed by the very extensive series of comparisons recently made at South ampton (see Comparisons of Standard of Length oj England, France, Belgium, Pmssia, Russia, India, and Australia, made at the Ordnance Survey Office, South ampton, 1866, and a paper in the Philosophical Transac tions for 1873, by Lieut. -Col. A. K. Clarke, C.B., R.E., on the further comparisons of the standards of Austria, Spain, the United States, Cape of Good Hope, and Russia). These direct comparisons, which were carried out with the highest attainable precision, are of very great value. The length of the toise has three independent determinations, viz., through the Russian standard double toise, the Prussian toise, arid the Belgium toise, giving for the length of the

toise, expressed in terms of the standard yard of England