357 



GEODESY. 



GEODESY. 



353 



of A, B, and have been observed, and which has itself been observed 

 from those point*. These observations give five independent angles 

 to be corrected, namely, two at p, and one at each of the other stations, 

 and three equations of condition, which are thus found : the two 

 triangles A P c and B p c give two equations of the first kind, namely, 



2, BPC + PBC + BCP=180 + E'. 



On considering the three triangles APB, BPO, ABC, it will be seen that 

 the side PB is common to -the first and second, BC to the second and 

 third, and AB to the third and first. Forming therefore the identical 

 equation 



._ P_B BC . AB 



AB I'B BC 



and substituting for those ratios those of the sines of the opposite 

 angles (each diminished by one-third of the spherical excess), we get 

 the equation of the second kind : 



3, 1 = 



BUI PAB SinBPC Sin ACB 



sin APB sin BCP sin CAB 



When the equations of condition have been thus formed, the 

 observed values of the different angles are substituted in them, each 

 being increased or diminished by a small indeterminate correction. 

 The values of the corrections are then determined simultaneously by 

 solving the equations according to the method of minimum squares, or 

 so that the equations of condition shall be satisfied as nearly as possible 

 (they cannot be all satisfied exactly), and the sum of the squares of 

 the corrections shall be a minimum. For further details on this 

 subject, and examples of the application of the theory to trigono- 

 metrical surveys, we must content ourselves with a reference to the 

 ' Supplementum Theoriae C'ombinationis,' &c. of Gauss (Gbttingen, 

 1828), where it is applied to a portion of the triangles surveyed by 

 Krayenhoff in Holland; to Nos. 121 and 122 of the ' Astron. 

 Nachrichten,' where it is applied by Rosenberger to Maupertuis's 

 measurement in Sweden ; to No. 438 of the same work, where it is 

 applied by Bessel to the computation of the triangles at the southern 

 extremity of the French arc of meridian : and to the ' Gradmestmiig in 

 Ost-1'reussen,' already referred to. The advantages of the method are 

 two-told. In the first place there is the probability that the result is 

 nearer the truth than if it had been deduced in any other way ; and 

 secondly, a general and uniform process of calculation ia substituted 

 for an imperfect and arbitrary one. 



In proceeding according to the ordinary method, and regarding the 

 triangles as independent of each other, the process is much simpler. 

 The difference between the sum of the three observed angles and 

 180 -t-E is the aggregate error of the three determinations. If each 

 angle was determined by an equal number of equally good observations, 

 the probable error would be the same for each, and the correction would 

 be properly made by dividing the aggregate error equally among the 

 three. If the observations are assumed to be equally good, but each 

 of the angles has been determined by a different number, then the 

 portion of the aggregate error which should be thrown upon each 

 angle is reciprocally proportional to the number of observations by 

 which it was determined ; but when the individual observations are 

 not equally good (and this is the general case) the distribution of the 

 aggregate error should be made in such a manner that the amount of 

 the correction to be applied to each angle is proportional directly to 

 the sum of the squares of the differences between each observation and 

 the arithmetical mean of the whole, and inversely as the square of the 

 number of observations by which the angle was determined. If an 

 angle has been determined by a single reading, the portion of the 

 aggregate error to be assigned to it may be made proportional to the 

 mean H/uare (that is, the sum of the squares of the differences from the 

 mean divided by the number of observations) of the errors of a series 

 of observations at one of the other angles made under similar atmos- 

 pheric circumstances. Such is the method which the theory of pro- 

 bable errors indicates ; but in most of the geodetical surveys which 

 have yet been published, the distribution of the aggregate error among 

 the three angles has been made, as already remarked, according to some 

 arbitrary hypothesis. 



The three angles of the triangle, corrected in the manner now de- 

 icribed, are regarded as the true geodetical angles, or rather as the 

 spherical angles formed by the arcs of the great circles which intersect 

 in the verticals passing through the stations on the surface of the 

 osculating sphere. In strictness there are no practical means of 

 determining the true geodetical angles, that is, the angles made by the 

 shortest lines on the spheroid. The observed angle is not the geodetical 

 angle, but the angle made by the two planes which intersect in the 

 vertical of the station, and pass through the remote signals. 



Calculatiim of the Sidet. The method of computing the sides which 

 first suggests itself is, to convert the given or known side into degrees 

 of a circular arc, whose radius is equal to that of the earth, and apply 

 the formula) of spherical trigonometry. This method has been some- 

 times adopted ; but as it gives rise to tedious calculations, it is usual 

 to have recourse to more expeditious processes, which, though only 

 approximative, give equally exact result*. Various methods of approxi- 



mation have been proposed, though there are only two which have 

 been much used. One of them is the method which has been exclu- 

 sively followed in the Ordnance survey, so far as published, and also, 

 generally by Delambre. It consists in deducing from the spherical 

 angles the corresponding angles formed by the chords, and then com- 

 puting the triangle by plane trigonometry. In this manner the chorda 

 of the two unknown sides are found, from which the sides themselves 

 are easily deduced. The formulae are as follows : 



Let the three spherical angles (that is, the observed angles corrected 

 For the errors of observation) be denoted by A, B, c ; the sides respec- 

 tively opposite by a, b, c (expressed in feet) ; and the radius of curva- 

 ture of the surface by r (also in feet). Let A' be the angle formed by 

 the chords of the sides 6 and c, and suppose A A'=Z seconds , ,ben 

 putting o>=206264'8", 



/Z>-e\ 



^-Tohn cot * A - 



In like manner if B' and c' denote the chord angles corresponding to 

 the spherical angles B, c, respectively, and if we suppose B B'=X' and 

 C o' = x", the two small corrections x' and x" will be computed from 

 similar expressions to the above, and we have then the three plane 

 angles A', B', c', the sum of which is 180. For computing these cor- 

 rections approximate values of thn sides must be previously found ; but 

 for this purpose it will generally be sufficient to use logarithms to four 

 decimal places.; 



Although for facility of explanation we have described the reduction 

 to the chord angles as applied to the corrected spherical angles, it 

 is manifest that it may be (in practice it generally is) applied to the 

 observed angles. In this case we get A' + B' + c' = 1 80 the aggregate 

 error of the three observed angles, which error must be then distri- 

 buted among the three reduced or chord angles in the manner before 

 described,- so that their sum may be exactly 180. By this means, 

 since we have obviously x + of + x" = E, the previous calculation of the 

 spherical excess, in order to correct the observations, is rendered 

 unnecessary. 



When the chords have been computed from the reduced angles A' 

 B', c', the arcs are found in terms of the chords by a well-known series, 

 of which it is only necessary to use the two first terms. Let a be a 

 small arc, and a' its chord, then o = a' (a /s -=- 24r 3 ). 



The other method of computing the sides to which we have alluded 

 depends on the following theorem, which was first given by Legendre. 

 If from each of the angles of a spherical (or spheroidical) triangle, the 

 sides of which are small in comparison of the radius, one-third of the 

 spherical excess be deducted, the sines of the angles thus diminished 

 will be proportional to the lengths of the opposite sides, so that the 

 triangle may be computed as in plane trigonometry. As before, let A 

 B, be the corrected spherical angles, a, 6, c the sides respectively 

 opposite, and E the spherical excess ; then, if we make 



A' = A-iE, B' = B-jE, c' = o-jE, 

 we shall have, in virtue of the theorem, 



, a sin B' a ein o' 



o == * > 



sin A' ' sin A' ' 



from which formulas the sides b and c are computed. This method 

 therefore requires no greater amount of calculation than would be 

 necessary if the triangles were on a plane surface, excepting that of tha 

 spherical excess ; and if the three angles are assumed to be determined 

 with equal accuracy, even this is not wanted (unless for the purpose of 

 testing the accuracy of the observations), the angles for calculations 

 being found at once by applying to each of the three observed angles a 

 third of the difference between their sum and 180. This is the 

 method which is most frequently adopted. 



Legendre's theorem will give a sufficiently accurate result in ordinary 

 cases ; but if the triangles are very large, and the utmost precision is 

 aimed at, it will sometimes be desirable to have a closer approximation. 

 This may be obtained by computing the angles A', B , o' from the 

 following expressions of their values, which appear to have been first 

 given by Professor Buzengeiger, in Lindenau's ' Zeitschrift fur Astro- 

 nomie,' vol. vi. (Tubingen, 1818), and which are equivalent to an ex- 

 tension of the theorem so as to include terms of the second order, 

 Legendre's approximation including only those of the first. As before, 

 let s be the area of the triangle, and < = 206264"'8, then 



l-A'= 5-1 + 



aa + 7bb + 7c<:\ 

 120rr /' 



7aa + bb + 7i 



120rr 



7aa + 7ii + ec\ 

 ~120rr /' 



and in consequence of these, the spherical excess E(=A + B + 180) 

 becomes 



us 



24rr 

 It is easy to see that the second terms of these expressions must 



