a—ia

72

The value of 2 (ft — d) is zero for any point on the arc ESF. and since the radius r of the small circle is not great the tangent at S of the arc ASH will not differ sensibly from the arc ESF within the small circle. Therefore from (I) for any point on ESF, =j sin (a + <j> 0 ) = — g sin (a — <t> ) /I', + DA ••• tftn * = (l),- D 1) tma , i /I>i + DA i ~.--, or r/> 0 = tan -1 - I ~~- - I tan a ■ U'->) where cf> o is the angle ESD, Therefore where <p is less than <j>, or greater than rr + <j> the error 2(ft —o) will be positive, and for all values of </!> between <ji 0 ami -rr + q!> 0 it will be negative. In other words, 2/3 is greater or less than 2a according as the point C lies above or below ESF. Suppose the point C to be contained in an elemental area dx by Ay, of which dx is in the direction of the radius anil dy at right angles, and let the angle subtended by dy at She d<f,. The number of such points as C within the semicircle above ESF, in which 2 (ft — a) is positive, is,— g-(» . r . */dx, dy) (16) i Id) , ~ Within the elemental area at a distance x from the centre there are X -j— such points as O. Therefore the sum of the angular errors for all points in the semicircle above ESF is given by the definite integral— f<i>o f r ■- (sm ( a + <£) i sin ( a z J+ o +*.J 0 \ i* d, m and the mean angular centring error is obtained by dividing this result by (16), and is---2 />, /"'^f si »(; + + t, r. ■ \ I), 1), / _4 . r // 1 1 2 cos 2a\ 3.77 V 1.)/ " DjD, / A similar expression with the negative sign determines the mean error of centring of the semicircle below ESF: hence the mean centring error is— E„ = + „I \'(T)S + I),'' - 21), IX cos 2a)/D, IX 3-r Now, A (D, 2 + D,' 2 - 21), 1)„ cos 2,0 Al5, therefore, jb c = + . , (17) 3-r. AS. I>S The distance between two stations adjacent to the one under consideration is termed by Mr. Briggs a transeotor: Thus Al> is the transector of the station S. Part 2. Limits of Ebeoes in Tbiangulation, In practice the accuracy of triangulation is judged from the measured and calculated length of a check base, or the agreement of the computed values of any line through a geometrical figure. A triangulation servos as a back-bone survey, the purpose of it being to establish a number of fixed points over the country with a high degree of accuracy. These triangulation stations control the traverses connected with them, and when well placed the traverses will usually run from one station to the next. The accuracy in the triangulation should be such that when the closing error in the traverse is being assessed there should be no need to take the triangulation error into account. Now, if x denote the error due to the traverse, and y the error in the triangulation, the total error zis represented by z— + VV 2 + if. If yin the above expression is one-third of x, z= + \ —- 1 05,t, or the influence of // on : is only about + 6 of x, and can be considered negligible. If the error in the traverse has a maximum value of 1 in (1,000, a suitable error in the triangulation should be 1 in 18,000. The triangulation error (a) can be divided into the error in the measurement of the baseline (/)) and the error due to the angular measurements (c), and therefore a = + Vb + c. Error in. Ease-line Measurement. With the steel ribband it is not difficult to measure a base-lino with an accuracy of I in 200,000, and the discovery of invar and its application to the measurement of distances provides a rapid and inexpensive method of base-line measurement of a high degree of accuracy. The metal invar is an iron-nickel alloy containing about 36 per cent, nickel. The coefficient of expansion of invar wires drawn in the steel-works of Imphy ranges from one hundred to fifty times less than that of steel.