01 iDBBT. 



QE IDB8T. 



M well as by direct observation. Thia circumstance permit* the obser- 

 vations to be nude in various ways, and afford* an important mean* of 

 rerifioation ; but in order that full advantage may be derived from it, 

 the obeerrationt miut be made and combined according to tome sys- 

 tematic plan. Strove, in the Rusuau triangulation. adopted the plan 

 of observing nioeaniTely the direction of every signal viaibl* from hi* 

 station, in reference to a certain arbitrary direction ; and the tame 

 method wat followed by Bessel. Thu appear* to be the mode of con- 

 ducting the observations by which an observer is enabled to make the 

 most of his position. 



Mtdmetum to tJu Cntrt of A Station. It is desirable that the centre 

 of the instrument should always be placed in the vertical line which 

 coincides with the axis of the signal at the same station ; but the Htrict 

 fulfilment of this condition may sometimes be impossible, or at least 

 extremely inconvenient. In such cases the instrument u placed near 



the station, and a correction made for the excentricity. Let c be the 

 centre of the station, E the place where the instrument is placed, A and 

 B the distant signals, so that A C B is the angle which U required, and 

 A E B the angle actually measured. Let the distance c E be denoted by 

 d, and A o (computed approximately) by m ; then the difference, in 

 second*, between ACS and A B is found from this formula 



ACB AEB = W d sin (BAC BBc)-t-m sin BAC, 



where = 206264 S", the number of seconds in an arc equal to the 

 radius, or .=cosec 1" =l-rsin 1". 



Reduction to tkc donjon. Although the theodolite has now come 

 into general, perhaps universal use, in carrying on important geodetical 

 operations, we shall add the formula by which an angle measured in 

 the oblique plane passing through the instrument and the two observed 

 objects is reduced to the horizon. Let A and B be the remote signals 

 C the angle in the oblique plane, c' its projection on the plane of the 

 horizon ; and let a and ft denote respectively the number of seconds by 

 which A and B are observed to be elevated or depressed above or below 

 the horizon of c ; then 



</= i J (a + 0) 1 tan I 0\ (a ft) 1 cot 4 c j sin 1*. 



When the angles have been measured, and (if necessary) deduced to 

 the centra of the station and the horizon, their values as given by the 

 instrument, being all affected with some portion of error which it is 

 impossible by any means to get rid of, must undergo a process of cor 

 rection or adjustment, or be made to satisfy certain mathematics 

 conditions, before a determinate result con be deduced from them. In 

 order to establish these conditions the following quantity must be com 

 puled for every triangle in the series. 



Spherical JSzeat. The spherical excess of a triangle on the surface 

 of a sphere or spheroid, formed by the sections of planes perpendicula 

 to the surfaces, is the excess of the sum of its three angles over 180 J 

 This excess has a given relation to the area of the triangle depending 

 upon the radius of the sphere ; and in a geodetical survey the dat< 

 for computing it are, in every case, a side c, and the three observe) 

 angles A,B, c, of which c is supposed opposite to e. Let 8 denote 

 the number of square feet in the surface or area of the triangle, R th 

 spherical excess in seconds, and r the radius of curvature in feet 

 we have then 8 = 4 cc sin A sin B -r sin o, and E = s-j-rr, wher 

 .=2062648". 



On the spheroid, the radius of curvature of a section perpendicular 

 to the surface is variable, and depends both on the latitude and th 

 inclination of the section to the meridian. For a series of triangles 

 included between two parallel* of latitude whose distance is not more 

 than two or three degrees, r may be supposed constant in computing 

 the spherical excess; and at the nearest approximation to it* mean 

 value, we may take the radius of curvature of the section which inter- 

 sect* the meridian in an angle of 45", at the middle latitude. The 

 general formula for the radius of curvature of an oblique section is 



eciuvi) 



hence also 



rom which l-=-r in readily computed for any value of 9. When 

 -- 45', the second terra vanishes. 



In the latitude of Greenwich (81 28' 89"), f - 20911981 feet, 

 (' = 20966473 feet, whence (making =45) log (,-j.rr) = 67251 10, 

 and consequently log E = log 8 + 0-67251 10. In order that E may 

 amount to 1', we must have log 8 = 9-39749, or 8 = 2,497,800,000 

 square feet, or nearly 76$ square miles, that is to say, the spherical 

 excess amounts to 1* for every 76{ square miles in the area of the 

 triangle. The calculation of the area therefore does not require to be 

 made with much accuracy, and may be facilitated by means of sub- 

 sidiary tables. 



Correction of the Otntrvalimi. One of the improvements for which 

 practical geodexy has recently been indebted to the Oerman astro- 

 lomers, particularly Gauss and Bessel, is a general method of combining 

 and correcting the observations according to the principles of the 

 theory of probability, *o as to elicit the result which is moit probably 

 nearest the truth, or which gives the nearest representation of the 

 whole of the observations. Formerly the practice was to regard each 

 triangle as a complete and independent whole, and to adjust the 

 observed angles (usually by some arbitrary process, or according to 

 the observer's judgment of their relative goodness), so as to fulfil the 

 condition of their sum being equal to 180 together with the spherical 

 excess, without regard to the relations subsisting among the angles of 

 the quadrilaterals or other polygons formed by the lines connecting the 

 angular points. But this mode of proceeding affords a very imperfect 

 solution of the problem : for in order to obtain the best result which 

 can be deduced from the observations, it is indispensable to have 

 regard not only to the condition just named, but to every independent 

 relation subsisting among the angles of the whole series of triangle* 

 included in the survey; and the more numerous the relations are 

 which the observations are made to satisfy, the greater will be the 

 probable accuracy of the final result. 



The equations of condition which express the independent relation* 

 connecting the angles of a system of geodetical triangles arise chiefly 

 from three sources : 1, The sum of the three angles of each triangle U 

 equal to 180 plut the spherical excess; which excess, being in all 

 case* a very small quantity, can be computed to so great a degree of 

 accuracy, that it may be regarded as absolutely exact. 2, If there be 

 a system of triangles so connected that the second has a side a in 

 common with the first, the third a side 4 in common with the second, 

 the fourth a side c in common with the third, and so on to the hut, 

 which has a side k in common with the preceding one, and another 

 ride / in common with the first, then, on forming the identical 



equation 



abed I 



1= 1 x a x 6 x c'"- x I-' 



and substituting for these ratios those of the sines of the angle* 

 opposite the respective sides, each being diminished by a third of the 

 spherical excess of the triangle to which it belongs, an equation ol 

 condition is obtained which should be satisfied by the observed anglei 

 3 When the angles observed at any station include the whole circuit 

 of the horizon, their sum must be egual to 860* ; but this condition 

 can only be made available when the angles are determined inde- 

 pendently of each other. 



In a complicated series of triangles, some difficulty may be found 

 in determining the exact number of independent relations furnished 

 by the angles and sides of the figures, but this will be materially 

 lessened by attention to the following considerations : If a point, p, 

 whose position is still unknown, be observed from two other point*, 

 A and B already determinud, and the directions of A and B be also 

 observed from r, we have then three angles for correction, and 

 one equation of condition of the first kind. If the unknown point,?, 

 be observed from three known points, A, B, c, and each of these be also 

 observed from P, wo have then five angles for correction, and three 

 equations of condition, namely, two of the first kind, furnished by tl 

 two triangles whose vertices are AP, and one of the second: 

 generally, when a point r has been observed from m stations whose 

 positions are already known, and each of these has been observe 

 from r, we liave then 2-l angles for correction (one at each 

 riven points and m-1 at t), and 2m- 3 independent equations of 

 condition, namely, w-1 of the first kind, and m-2 of the secoi 



An example will render this sufficiently clear. Let A, B, o U 



where a is the radius of the equator, e the excentricity (so that 

 ee=(aaU>)^-aa, b being half the polar axis), I the latitude, and 6 the 

 azimuth, or inclination of the section to the meridian. 



Let f denote the radius of curvature of the meridian (for which 

 =0), ' that of the section perpendicular to the meridian (for 

 which t^W), the formula gives 



1^ -/(!- sin'O 1 |1 V(l-*in'Q 

 P * a(l-) ; ? " a 



point* already determined, and P a new *tation at which th directions 



