S4X = 63A. X . tanA S4L = +54A. Y.tanA S,A = +S4L .Z
The calculations described so far suffice to make the angles of the several trigonometrical figures consistent inter se, and to give preliminary values of the lengths and azimuths of the sides and the latitudes and longitudes of the stations. Reduction The results are amply sufficient for the requirements of Principal
of the topographer and land surveyor, and they are published in preliminary charts, which give full numerical tion.
details of latitude, longitude, azimuth and sidelength, and of height also, for each portion of the triangulation—secondary as well as principal—as executed year by year. But on the completion of the several chains of triangles further reductions became necessary, to make the triangulation everywhere consistent inter se and with the verificatory baselines, so that the lengths and azimuths of common sides and the latitudes and longitudes of common stations should be identical at the junctions of chains and that the measured and computed lengths of the baselines should also be identical.
As an illustration of the problem for treatment, suppose a combination of three meridional and two longitudinal chains comprising seventytwo single triangles with a baseline at each corner as shown in the accompanying c e diagram (fig. 2) ; suppose the
three angles of every triangle to have been measured and made consistent. Let A be the origin, with its latitude and longitude given, and also the length and azimuth of the adjoining baseline. With these data processes of calculation are carried through p the triangulation to obtain the
lengths and azimuths of the FIG. 2.
sides and the latitudes and longitudes of the stations, say in the following order: from A through B to E, through F to E, through F to D, through F and E to C, and through F and D to C. Then there are two values of side, azimuth, latitude and longitude at E—one from the righthand chains via B, the other from the lefthand chains via F; similarly there are two sets of values at C; and each of the baselines at B, C and D has a calculated as well as a measured value. Thus eleven absolute errors are presented for dispersion over the triangulation by the application of the most appropriate correction to each angle, and, as a preliminary to the determination of these corrections, equations must be constructed between each of the absolute errors and the unknown errors of the angles from which they originated. For this purpose assume X to be the angle opposite the flank side of any triangle, and Y and Z the angles opposite the sides of continuation; also let x, y and z be the most probable values of the errors of the angles which will satisfy the given equations of condition. Then each equation may be expressed in the form [ax+by+cz] =E, the brackets indicating a summation for all the triangles involved. We have first to ascertain the values of the coefficients a, b and c of the unknown quantities. They are readily found for the side equations on the circuits and between the baselines, for x does not enter them, but only y and z, with coefficients which are the cotangents of Y and Z, so that these equations aresimply[cot Y.y—cot 'Z.z] E. But three out of four of the circuit equations are geodetic, corresponding to the closing errors in latitude, longitude and azimuth, and in them the coefficients are very complicated. They are obtained as follows. The first term of each of the three expressions for &A, AL, and B is differentiated in terms of c and A, giving
d.DA= DA do—dA tan A sin i"
d.AL= DL do+dA cot A sin i" (7)
dB=dA+AA 1 c+dA cot A sin 1"
AL" =
AA" or
B—(ir+A)=
(5)
End of Article: S4X 

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