AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 875 



fraction. This is proved true only in part (p. 872). It is shown that such a ratio is of 

 value in assessing the probable accuracy of the individual lines of the survey (p. 867), but 

 not necessarily in determining the accuracy of the triangulation in distance-transmission. 

 This latter point is further discussed below. It will also be evident that if the main 

 and verification bases are measured by the same steel-tape, the latter will provide no 

 check on cumulative errors in measurement, such as that resulting from the tape being- 

 incorrect in length. 



(g) In a triangulation scheme covering a certain area of ground, the fewer the 

 triangles the better (40). The importance of this fact is emphasised. 



(h) It is permissible to use triangles towards the end of a scheme departing more 

 from the perfect shape than those near the beginning. 



(B) (a) Though essentially a mathematical theorem, the mode of summing vector 

 errors (l) perhaps deserves first place in this series of new results, since much of the 

 work — particularly the study of the accuracy of triangulation in distance-transmission 

 — would have been very difficult, if not impossible, without it. 



(b) A method of assessing the average error due to imperfect centring is given 

 (Section II.), allowing of this class of angular error being studied in relation to that due 

 to imperfect sighting and reading of a surveying instrument (Section III.). 



(c) It is shown that the average error due to centring depends on the value of the 

 traverse-angle, being a maximum when that angle is 180° (6). 



(d) Formulae are derived [(12), (15), and (16)] which allow of the average total 

 error being computed in any traverse, whether "closed" or " open," and whether run 

 by a theodolite or compass instrument. By their application it is possible to determine 

 if the error of closure in a " closed " traverse is less or greater than the average, and 

 thus they provide a criterion of accuracy more satisfactory than that given by express- 

 ing the closing error as a fraction of the total length of the traverse. The chief objection 

 to the formulae is their cumbrousness, and the apparent impossibility of reducing their 

 size by any process of approximation without at the same time nullifying any value 

 they may possess. Trial will show, however, that computation by means of them is not 

 such a lengthy process as would at first seem (providing tables of squares, square-roots, 

 and reciprocals are at hand), since round numbers only need be employed for the lengths 

 and angles. In obtaining the average value of the total error of a theodolite traverse, 

 (12) is first used, and the results afterwards substituted in (16). 



Perhaps the chief utility of these results lies in the fact that they enable the error 

 which might reasonably be expected in any " open " traverse to be ascertained. 



(e) By formulae (17) and (18) the average error respectively in the total latitude and 

 total departure of the end point of a traverse can be computed. These formulae are of 

 use when it is desired to find the error likely to occur in any given direction, one co- 

 ordinate axis being, for the purpose of the calculation, arranged in that direction. For 

 instance, in the well-known problem in practical surveying to determine the length and 

 bearing of the closing side of a polygon, the average error in both these unknown 



