Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/621

599 EARTH 509 tion whereby a correction for temperature can be applied to the bars so as to reduce their length to that at the standard temperature. The bars in measuring were not allowed to come into contact, but the intervals left were measured by the interposition of a glass wedge. The results of all the comparisons of the four measuring rods with one another, and with the standards, are elaborately worked out by least squares. The angles ware observed with theodolites of 12 and 15 inches diameter, and ths latitudes determined by means of the transit instrument in the prime vertical a method much used in Germany. The formulae employed in the reduction of the astronomical observations are very elegant. The reduction of the triangulation was carried out in the most thorough manner, the sum of the squares of all the actual theo dolite observations being made a minimum. As it is usual now to follow this method (sometimes only approxi mately) in all triangulations where great precision is required, we here give a brief description of the method. The equations of condition of a triangulation are those which exist between the supernumerary observed quantities and their calculated values, for, after there are just sufficient observations to fix all the points, then any angle that may be subsequently observed can be compared with its calculated value. If a triangulation consist of n + 2 points, two of which are the ends of a base line, then to fix the n points 2?i angles suffice ; so that if m be the actual number of angles really observed, the triangulation must afford m - 2/i equations of condition. To show how these arise, suppose that from a number m of fixed points A, B, C. . .a new point P is observed, which m points are again observed from P, then there will be formed in - 1 triangles, in each of which the sum of the observed angles is = 180 + the spherical excess; this gives at once m - 1 equations of condition. The m ~ 2 distances will each afford an equation of the form _TC PB_ PA _ PB PA PC ~ not, however, limited to three factors. Should P observe the m points and not be observed back, there will be m 3 equations of the above form (they are called side equations). In a similar manner other cases can be treated. In practice the ratios of sides are replaced by the ratios of the sines of the corresponding opposite angles. To each observed angle a symbolical correction is applied, so that if a be an observed angle and a + x the true or most probable angle, sin (a + x) = sin a(l +x cot a), x being a small angle whose square is neglected. Thus the side equation takes the form fi + /3 l x l + (3. 2 x. 2 + . . . (3 r x r = Q. In the case of equations formed by adding together the three observed angles of a triangle the co-efficients are of course unity. The problem then is this : Given n equations between m(m&amp;gt;?i) unknown quantities x v. . ,x m) which are the corrections (expressed in seconds of arc) to the observed angles, it is required to determine these quantities so as to render the function w l x 1 2 + w^x^ + w^+ . . . w m x m 2 a minimum, where w . . . -w m are the weights of the deter minations of the angles to which the corresponding correc tions belong. The corrections x l ., . x m fulfilling this condition of minimum have, according to the theory of least squares, a higher probability than any other system of corrections teat merely satisfy the equations of condition. Multiply the n equations by multipliers l} X 2 , . . . X n , and we obtain by the theory of maxima and minima m equa tions = (8mA.! + + m &quot; 3 +. . The values of x i. . . # m obtained from these equations are to be substituted in the original equations of condition, and then there will be n equations between the 11 multipliers X x. . . X n. These being solved, the numerical values of X 1. . . X n will be obtained, and on substituting these in the last equations written down, the values of x l. . . x m will follow. The process is a long and tedious one; but it is inevitable if we wish very good results. The great meridian arc in India was commenced by Colonel Lambton at Punnce in latitude 8 9. Follow ing generally the methods of the English survey, he carried his triangulation as far north as 20 30. The work then passed into the able hands of Sir George (then Captain) Everest, who continued it to the latitude of 29 30. Two admirably written volumes by Sir George Everest, published in 1830 and in 1847, give all the details of the vast under taking. The great trigonometrical survey of India is now being prosecuted with great scientific skill by Colonel Walker, R.E., and it may be expected that we shall soon have some valuable contributions to the great problem of geodesy. The working out of the Indian chains of triangle by the method of least squares presents peculiar difficul ties, but enormous in extent as the work is, it is being thoroughly carried out. The ten base lines on which the survey depends were measured with Colby s compensation bars. These compensation bars were also used by Sir Thomaa Maclear in the measurement of the base line in his exten sion of Lacaille s arc at the Cape. The account of this operation will be found in a volume entitled Verification and Extension of Lacaille s Arc of Meridian at the Cape of Good Hope, by Sir Thomas Maclear, published in 1866. Lacaille s amplitude is verified, but not his terrestrial measurement. The number of stations in the principal triangulation of Great Britain and Ireland is about 250. At 32 of these the latitudes were determined with Ramsden s and Airy s zenith sectors. The theodolites used for this work were, in addition to the two great theodolites of Ramsden which were used by General Roy and Captain Kater (and which are now in as good condition as when they came from the hands of the maker), a smaller theodolite of 18 inches diameter by the same mechanician, and another of 24 inches diameter by Messrs Troughton and Simms. Observations for determination of absolute azimuth were made with these instruments at a large number of stations ; the stars a, 8, and X Ursse Minoris and 51 Cephei being those observed, always at the greatest azimuths. At six of these stations the probable error of the result is under 0&quot; 4, at twelve under 0&quot; 5, at thirty-four under 0&quot; - 7 : so that the absolute azimuth of the whole network is determined with extreme accuracy. Of the seven base lines which have been measured, five were by means of steel chains and two with Colby s compensation bars. This is a system of six compound bars self-correcting for temperature. The compound bar may be thus described. Two bars, one of brass and the other of iron, are laid side by side, parallel, and firmly united at their centres, from which they are free to expand or contract ; at the standard temperature they are of the same length. Let AB be one bar, A B the other ; draw a line through the corresponding extremities A, A to P, and a line through the other extremities B, B to Q, make A P = B Q, AA being = BB. Now if A P is to AP as the rate of expansion of the bur A B to the rate of expansion of the bar AB, then clearly the distance PQ will be invariable, or very nearly so. In