361 



GEODESY. 



GEODESY. 



362 



small differences, 100 feet for "example. In following this method, 

 however, the stations must not be so far from the assumed meridian 

 that the difference of the curvilinear distance and its projection on the 

 horizontal plane becomes sensible, and hence in the survey of a large 

 country the direction of a new meridian (or the azimuths) requires to 

 be determined astronomically when the triangulation has been carried 

 a degree or two to the east or west of the point of departure. It is 

 scarcely necessary to add that the positions of the secondary points 

 are computed without reference to the curvature, or convergence of 

 meridians. 



The method of computing the distance between the parallels of two 

 remote stations, A and F, which we have now described, is that which 



along the ~ ------ -------- , 



the line to be measured are not so great as to give rise to any error by 

 reason of the neglect of the curvature of the perpendicular arc. For 

 other methods of computing the parts of the meridian, and formula;, 

 theoretically more exact, for computing the latitudes, longitudes, and 

 azimuths, see Delambre's ' McSthodes Analytiques pour la Determina- 

 tion d'un Arc du Mendien ; ' or Puissant, ' Trait<5 de GiSodesie,' third 

 edition, 1842. 



If the earth were a regular spheroid of rotation, the latitudes and 

 longitudes deduced from geodetical measurements commencing with a 

 given observatory would agree exactly (supposing no errors) with those 

 given by astronomical observations. Such agreement however is not 

 found to exist, and no regular figure can be assigned to the earth by 

 which the results of the two methods can be entirely reconciled. An 

 instance or two will suffice to give an idea of the extent to which the 

 discordances may reach. In the remeasurement of Beccaria's arc, the 

 astronomical difference of latitude between Andrate and Mondovi was 

 found to be 1 7' 26".98, while the difference computed from the French 

 triantmlation, taking the observatory of Paris as the point of departure, 

 and assuming the ellipticity = 1 -=- 308'65, was 1 8' H".82. The dis- 

 cordance amounts to 47".84, corresponding to a distance on the meri- 

 dian of about 4880 English feet, the whole distance being about 

 368,885 feet. Again, the latitude of Venice deduced from that of 

 Rimini by triangulation was found to differ 17".2 from the latitude 

 given by direct observation ; and the latitude of Rimini deduced from 

 that of Milan differed 27".4 from the astronomical latitude. (' Con- 

 naissance des Terns," 1827.) Similar anomalies have been found in 

 the surveys in England, France, Austria, and indeed all other countries ; 

 and as their amount exceeds that which can with any probability be 

 assigned to errors of observation, they are ascribed to irregularities in 

 the direction of gravity arising from inequalities in the form or internal 

 structure of the earth, to the attraction of mountains or local varia- 

 tions of density. Puissant (' M<Sm de 1'Acad.,' t. xiv., 1838) shows 

 reason for supposing that in France the curvature of the surface is 

 considerably different on the east and west sides of the meridian of 



I'.Ulrf. 



Determination of the Altituda.ln order to complete the description 

 of the objects embraced in a trigonometrical survey, it only remains for 

 us to point out the manner in which the relative heights of the stations 

 are observed and computed. The observations required for this pur- 

 pose are the zenith distances of the signals as seen from each other ; 

 and they may be made with the theodolite or any instrument with 

 which angles can be measured in a vertical plane. The chief difficulty 

 attending the determination arises from the uncertainty of the ter- 

 restrial refraction, an element which is liable to frequent and con- 

 siderable variations, more especially as the objects observed are always 

 nearer the horizon. The formula: are as follows : 



Let A and h' denote respectively the heights of two stations, A and B, 

 above the level of the sea ; z the zenith distance of B as observed at A ; 

 Az the seconds of arc through which B is elevated by the refraction ; 

 /and Az 7 , the corresponding quantities in respect of A as observed at B; 

 c the angle at the centre of the earth formed by the verticals of A and 

 B ; d the distance in feet between A and B on the spheroid ; and r the 

 radius of curvature of the terrestrial arc. The plane triangle formed 

 by the two verticals and the straight line which joins A and B gives 

 this analogy : 



A: h' K ::cotlo:tan J {z' + Az* (z + Az)}, 



from which, on rejecting superfluous quantities, and with the aid of 

 certain physical assumptions, the formula: for computation are de- 

 duced. 



In the first place, /' + A may be rejected as being insensible in com- 



parison of 2r : then since z" + A/ (z + Az)=2 {90 (z + A2-Jc)},the 

 above analogy gives 



h'- h = 2r tan 4 c cot (z + Az- \c). 



Again, because c is always a very small angle, we may put, without 

 sensible error, d = 2r tan 4 c. The last formula then becomes 

 V A = d cot (z + Az 4 c). 



With respect to the unknown quantity Az, two assumptions are 

 made : first, it is assumed that the whole effect of the refraction at 

 both stations is proportional to the distance between the stations, or to 



the angle c; or to assume A/ + Az=io, where i is a numerical co- 

 efficient. Secondly, it is assumed that the effect of the refraction is 

 the same at both stations, or that As^Az; this gives Az=4ic, and 

 consequently Az 4 c = 4 (1 k) c. Substituting this in the last 

 equation, we get 



h'k=d cot {z J (1-i) c} (a). 



This equation gives the height of the signal at B above the place of 

 the instrument at A in terms of the geodetical distance d, and the 

 observed angle z, assuming the coefficient of refraction to be known. 

 It will be observed that c is given in seconds by the formula c = a> d-t-r, 

 where o> = 206264"-8. 



When the zenith distances are observed at both stations, the co- 

 efficient may be deduced from the observations ; for the assumption 

 of A2' + Az= c gives c=c-t-180 (z + z), whence 



l-fc = (z' + z-180 )-=-c. (b). 



A mean value of k, deduced in this manner from a number of reci- 

 procal observations, may be substituted in the equation (a) for finding 

 the difference of the heights of two stations when the zenith distance 

 has been observed at one of them only. But when z' and z are both 

 observed, the difference of altitude is obtained independently of the 

 value of k ; for on substituting the value now given of 1 -k in 

 equation (a), we get 



/ t '-ft=dtan4(z'-z). (c). 



The absolute height h of the first station is usually found by levelling 

 from the surface of the sea at half-tide. But assuming the refraction 

 to be known, the absolute height of a station may be determined by 

 observation of the zenith distance of the sea horizon. The formula is 



h = 4 c (1 + 4*) 2 tan' (z- 90). (d ). 



In the application of the preceding formula:, it is necessary to attend 

 to the height at which the instrument was placed with respect to the 

 signal, or point for which the calculation is to be made. For instance, 

 if the object observed is the surface of the ground, and the instru- 

 ment is placed at the height of n feet above the surface at both 

 stations, then on computing the coefficient of refraction from the 

 formula (6), the angles z' and z must be each diminished by the number 

 of seconds in the angle subtended by n feet at the distance d, that is, by 

 206264"'8 n-r-d. In the formula (c), which applies to reciprocal obser- 

 vations, the height of the instrument need not be regarded, provided it 

 be the same with respect to the signal at both stations. In the cases to 

 which (a) and (d) apply, the correction is made by subtracting the 

 height of the instrument above the ground from the results given 

 immediately by the formula;. 



The surest determination of altitudes is that which is given by 

 reciprocal observations ; for in this case the only assumption involved 

 in the formula is, that the refraction is the same at both stations ; and 

 if the observations are made under similar atmospheric circumstances, 

 this cannot well be supposed to lead to error. Such observations also 

 give a more certain value of the coefficient cf refraction than can be 

 deduced from the astronomical theory, which, besides the hypotheses 

 necessary for connecting the variation of temperature with the altitude, 

 assumes also (for the present purpose) that the variation of tempera- 

 ture follows the same law throughout the whole distance from the one 

 station to the other. The mean value of k, deduced by Bessel from 

 reciprocal observations made during the measurement of the Prussian 

 arc, was 0'1370 ; Gauss found 0'1306; Carabceuf, from the French 

 triaugulation in Piedmont, 0'1285 : Struve, in Russia, 0'1237/i De- 

 lambre and Me'chain, 0'1566 ; and in the English survey the definitive 

 value of k was found to be 0'0809 for rays crossing the sea, and 0'0750 

 for rays not crossing the sea. 



We shall conclude this article with a short statement of some of 

 the more important results relative to the figure and dimensions of the 

 earth which have been deduced in recent years from a discussion of 

 arcs of the meridian and of parallel. 



In the ' Encyclopedia Metropolitan;! ' (art. ' Figure of the Earth '), 

 Mr. Airy has discussed fourteen arcs of the meridian and four arcs of 

 parallel. The results obtained by him are 



0=20,923,713 feet. 

 6 = 20,853,810 



a-b _ 1 

 *~ a ~ 299-33 feet, 



a, It, denoting the semi-major and semi-minor axes of the terrestrial 

 spheroid, and < the ellipticity or compression. 



In the ' Astronomische Nachrichten,' Nos. 333 and 438, Bessel has 

 given an elaborate discussion of the figure and dimensions of the 

 earth, founded on ten arcs of the meridian.* His final results present 

 a remarkable accordance with those obtained by Mr. Airy. They i 



0=20,923,600 feet. 

 6=20,853,656 



f ~ 29905 ' 

 In 1841 



