AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 863 



triangles of secondary importance (secondary or tertiary triangles) that two angles only 

 are measured, leaving the third to be obtained by difference. 



In this section minor triangulations only are discussed ; in these the question of 

 ' ' spherical excess " does not enter. 



The errors of measurement in triangulation are of two kinds, namely : (a) Errors in 

 the angles, and (b) Error in the base. 



We are justified in taking the average error in the angles of any triangle of equal 

 amount, if they have been measured by the same method, and observer, and with the 

 same care, since the sides of the triangle are of such lengths as to render centring 

 errors negligible (see p. 857). Throughout this section the letter v is taken to 

 represent the average error in measurement of an angle, expressed in radians. 



A uniform method is adopted of representing linear errors : thus c x is the average 

 error in the side c of a triangle, d 1 in a side d, and so on. 



(1) The Case of a Single Triangle. 



Firstly, consider the case of a single triangle standing on a measured base — such a 

 triangle, for example, as ABC of fig. 4. The triangle forms one link in a chain of 

 triangles ; hence errors in measuring the angles or base will not only affect the calcu- 

 lated lengths of the unmeasured sides, but will be carried forward through the whole 

 scheme ; it is therefore necessary to deal with the triangle as an agent for transmitting 

 distance. 



A train of triangles may subsequently be built upon the side a, or upon b, or upon 

 both. In a general discussion, therefore, these sides must be considered as being of 

 equal importance ; in other words, they must be given an equal " weight." Hence, in 

 determining the best shape of the triangle, a must not be allowed to suffer for the 

 sake of b, nor b for the sake of a. This condition can only be realised when a and b 

 are kept equal in length ; that is to say, the best-conditioned triangle must be some 

 form of isosceles one. 



Now, a triangle will have the best shape when it fulfils its function as a distance- 

 transmitter with a minimum of error. Thus such a triangle as ABC, fig. 4, will be of 

 the best shape when the error in a (or in b) forms the smallest possible proportion of 



the length of a (or of b) ; i.e. when ( h = ^ is a minimum; for it is the fractional 



a b 



error -J or -^ which must be considered in its effect on the next triangle of the scheme. 



a b . ■ 



The triangle ABC being isosceles when of the best shape, we have — 



G=\80°-2A ..... (22) 



To calculate the length of the side a, the " sine rule " is used, thus — 



c sin A . 



a = — — pr ..... (23) 



sin C 



