AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 851 



Let the position of the point A, fig. 1, be affected by independent vector errors, 

 whose average magnitudes are respectively ±x x , and dzx 2 , and let a be the angle 

 between their respective directions of action. 



The diagram shows that the square of the sum of these errors is either 



R 2 = x\ + xi + 2* 1 ar 2 cos a 

 or 



R 2 = x'i + xl - 2x^ 2 cos a, 



Therefore, if we were to examine a considerable number of results, all affected by these 

 vector errors, we should find that the mean value of R 2 came very nearly equal to x\ + x\, 

 since the terms in cos a, being as often negative as positive, would only influence such 

 a mean to an extent negligible in comparison with that of the other two terms. Hence 



we may write — 



K--±J(xi + xl) (1) 



More than two vector errors can be similarly treated, and it is immaterial to the 

 result whether their directions lie in the same plane or not. 



Fig. 1. 



The theorem is a generalisation of the well-known principle to the effect that the 

 average error of the sum of a number of observations is equal to the square-root of the 

 sum of the squares of the average errors of each observation ; and it facilitates very 

 considerably the following discussion of the propagation of errors in surveying. 



Section II. The Average Error due to Imperfect Centring. 



If a displacement, r, be made in centring a theodolite over a traverse-station, the 

 instrument will actually be centred over some point on the circumference of a circle 

 whose radius is r. Let this small circle be represented by that in fig. 2, its centre being 

 0. Let P x and P 2 , the two traverse-stations adjacent to 0, be respectively at distances 

 L x and L 2 from it ; and let N be any point on the circumference of the circle. 



Then, if the instrument is actually centred over N, the angle P 1 NP 2 (or S) will be 

 measured in place of the angle P x OP 2 (or T) ; the angular error due to imperfect 

 centring will therefore be (S — T). The error has a plus sign for such a point as 

 N and a minus sign for such a point as Z ; therefore there are two points on the 

 circle, namely E and F, at which the angular error is zero. The latter points lie on 

 the circumference of a circle passing through P v 0, and P 2 , whose radius must always 

 be very large even when the stations P l and P 2 are close together ; hence the points E 

 and F may be considered as occupying exactly opposite positions on the circle NFZE. 



