AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 867 

 When the triangle is equilateral, equation (25) becomes — 



^y{f-&n <*> 



(2) The Case of a Chain of Triangles. 



Fig. 4 shows a chain of triangles dependent on the measured base, e. When the 

 triangles come to be solved a will serve as base for the second triangle, d for the third, 

 and so on. 



We have from (24), no matter what may be the shape of the triangles— 



j = ± yj \ w 2 (cot 2 D + cot 2 F) + (j 1 ) j- = ± J | « 2 (cot 2 D + cot 2 F + cot 2 C + cot 2 A) + (j 1 ) } , 



and if - be the average fractional error in a side of the nth triangle — 



h__ + IS ./Sum of the squares of the cotangents of all the angles\ , ( c j\ 2 I /r> 7 \ 



z ~ V I V of the scheme which are opposite internal sides * ) \c ) ) ^ u ' 



In practice, whether the survey is accurate or not is judged from a comparison of 

 the measured and calculated length of a check-base. 



If z is this check-base, placed at the extreme end of the scheme, and z A is its 

 calculated length, equation (27) provides a measure of the average fractional error in 

 that quantity. Let z B be the measured length of z, and z 2 the average error in 



measuring it. 



Then the difference between the calculated and measured lengths of z will be 

 due to Zj and z 2 combined ; hence if z 3 represent the average value of this difference, 

 z 3 = zh s /(z'l + zl), or, to state the average fractional difference — 



2 3_ + / 1 , sASum of the squares of the cotangents of all\ , f<h\ , / Z A" 1 , 98 \ 



z _ V 1 V angles opposite internal sides / v: / \z ) ) ^ ' 



When the triangles are roughly equilateral (28) reduces to — 



r-V^©'*®'} • ■ ■ ■ <-> 



When a check-base forms a side of the triangle most remote from the base it is 

 generally taken for granted that, if the measured and calculated values agree to, say, 1 

 in 12,000, the accuracy of the triangulation as a whole may be expressed by that ratio. 

 Towards the end of the paper some criticism on this assumption is attempted. There 

 can be no doubt, however, that the criterion is a most valuable one, and its utility in 

 one important respect falls to be mentioned here : — 



A minor triangulation is usually performed to serve as a backbone survey ; by 

 means of it a number of fixed points scattered over the property are established with 

 a high degree of accuracy. These points are afterwards linked together by traverses 

 which are for the purpose of gathering in detail. If the triangulation stations have 

 been well placed, there will never be need for these traverses to be extensive ; the 



* All such lines as a, d, g, etc. , which are not bounding lines of the area covered by the scheme, are spoken of here 

 as internal sides. 



