Page:Encyclopædia Britannica, Ninth Edition, v. 22.djvu/731

 TRIANGULATIOX.] SURVEYING 701 AA" or ~ sin A tan X cosec 1'' -. *-] l + 2tan 2 X + 4 v 1 I l-e 2 I sin 2A cosec 1' ^ K ^ J- ^ ; <*V5 .,, tan X . ? sin 3 A tan X (1 + 2 tan 5 X) cosec 1" 6 v 3 nil). } Each A is the sum of four terms symbolized by 5j, .,, 3 3, and 8 t ; the calculations are so arranged as to produce these terms in the order 5X, 5Z, and 8A, each term entering as a factor in calculating the following term. The arrangement is shown below in equations in which the symbols P, Q, . . . Z represent the factors which depend on the adopted geodetic constants, and vary with the latitude ; the logarithms of their numerical values are tabulated in the Auxiliary Tables to Facilitate the Calculations of the Indian Survey. .,= + 8 l A..smA.c = + S lt .Q.sec.ta.nA S l A = =-8.,.S.cotA 8.,A=+8 fi L.T = + 8 3 .U.smA.c 8 3 A = + o 3 L. By this artifice the calculations are rendered less laborious and made susceptible of being readily performed by any persons who are acquainted with the use of logarithm tables. Limits of 16. Limits within which Geodetic Formulae may be em- geodetic ployed without Sensible Error. Each A is expressed as a formula. ser j es O f ascending differentials in which all terms above the third order are neglected ; for the side length c in no case exceeded 70 miles, nor was the latitude ever higher than 36, and for these extreme values the maximum magni- tudes of the fourth differential are only 0"'002 in latitude and 0"'004 in longitude and azimuth. Far greater error may arise from uncertainties regard- ing the elements of the earth's figure, which was assumed to be spheroidal, with semi-axes a = 20, 922, 9 32 feet and 6 = 20,853,375 feet. The changes in AA, AZ, and A^4 which would arise from errors da and db in a and b are indicated by the following formulae : , . _ dp dv -.(dv ' p 2 ' 7~ 3 ~(l-e 2 )e. d.AL = - AZ. ^ - 5 2 Z.^ - (5 3 Z + S 4 L)2~ v idv dp p v p ) J A A A S A- d.AA = - A A. S^A 1 2tan 2 X + - P in which 2de ^=- -000,000, 0478 {da - 2db -S(da- db) sin 2 X} ^= + '000,000,0478 {da + (da- db) sin 2 X} 000, 0145 {da-db} ..(14). (1 - e-)e The adopted values of the semi-axes were determined by Colonel Everest in an investigation of the figure of the earth from such data as were available in 1826. Forty years afterwards an investigation was made by Captain (now Colonel) A. R. Clarke with additional data, which gave new values, both exceeding the former. 1 Accepting these as exact, the errors of the first values are da = - 3130 feet and db = - 1746 feet, the former being 150, the latter 84 millionth parts of the semi-axis. The corresponding changes in arcs of 1 of latitude and longitude, expressed in seconds of arc and in millionth parts (/JL) of arc-length, are as follows : Inlat. 5V.AX= -"'069 or ,, 15 ,, -"-113 ,,31,, ,, 25 -"-195,, 54,, 35 -"-303 ,,84,, = -"-540orl50/* ; -"-554 154,,; -"-581 ,,161,,; -"-617 ,,171,,. These assumed errors in the geodetic latitudes and longi- tudes are of service when comparisons are made between independent astronomical and geodetic determinations at 1 See A ccount of the Principal Triangidation of the Ordnance Sur- vey, 1858, and Comparisons of Standards of Length, 1866. any points for which both may be available : they indi- cate the extent to which differences may be attributable to errors in the adopted geodetic constants, as distinct from errors in the trigonometrical or the astronomical operations. 17. Final Reduction of Principal Triangulation. The Reduc- calculations described so far suffice to make the angles of tion f the several trigonometrical figures consistent inter se and P rinci ral to give preliminary values of the lengths and azimuths of * the sides and the latitudes and longitudes of the stations The results are amply sufficient for the requirements of the topographer and land surveyor, and they are published m preliminary charts, which give full numerical details of latitude, longitude, azimuth, and side-length, 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 reduc- tions became necessary, to make the triangulation every- where consistent inter se and with the verificatory base- lines, 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 base-lines should also be identical. How this was done will now be set forth. But first it must be noted that the triangulation might at the same time have been made consistent with any values of latitude, longitude, or azimuth which had been determined by astronomical observations at either of the trigonometrical stations. This, however, was undesirable, because such observations are liable to errors from deflexion of the plumb-line from the true normal under the influence of local attraction, and these errors are of a much greater magnitude than those that would be generated in triangu- lating between astronomical stations which are not a great distance apart. The trigonometrical elements could not be forced into accordance with the astronomical without altering the angles by amounts much larger than their probable errors, and the results would be useless for in- vestigations of the figure of the earth. The only inde- pendent facts of observation which could be legitimately combined with the angular adjustments were the base-lines, and all these were employed, while the several astronomical determinations of latitude, differential longitude, and azimuth were held in reserve for future geodetic investi- gations. As an illustration of the problem for treatment, suppose a com- Specific bination of three meridional and two longitudinal chains comprising illustra- seventy-two single triangles, with a base-line at each corner, as shown tion in the accompanying diagram (fig. 2) ; suppose the three angles of every triangle to have been c B measured and made con- sistent. Let A be the ori- gin, with its latitude and longitude given, and also the length and azimuth of the adjoining base-line. With these data processes of calculation are carried through the triangulation to obtain the lengths and_ AAAAAAAAAA/ > / E 1 F "
 * - - 3 { - + tan- X I sin 2^1 cos ^4 cosec 1

wwvwvvv azimuths of the sides and the latitudes and longi- S- tudes 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 right- hand chains r-ia B, the other from the left-hand chains via F ; similarly there are two sets of values at C ; and each of the base- lines 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