AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 855 



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angle so measured would be— 7^ x v3, or I 2 seconds, for the particular observer and 



instrument. 



Nor, if the surveyor carefully examine his own practice, is the displacement r difficult 

 to ascertain — at any rate approximately. Thus, if he were in the habit, in ordinary 

 traversing, of centring to within a quarter of an inch, he would disallow all displace- 

 ments greater than that amount, but accept, as being sufficiently near, any of a smaller 

 magnitude. Now, if it were true that there was an equal likelihood of the occurrence 

 of any deviation from centre between zero and a quarter of an inch, the average 

 deviation would be one-eighth of an inch ; yet, if the actual method of centring be borne 

 in mind — in which the instrument is moved until the point of the plumb-bob comes 

 within an imaginary circle, in this case a quarter of an inch in radius — it will be evident 

 that there is a greater chance of displacements approaching the maximum than of those 

 approaching zero. Probably by taking three-quarters (rather than half) of the maxi- 

 mum deviation a value for r will be obtained approximating closely to the true average. 

 In the present example this would give three-sixteenths of an inch as the average r. 



Having determined, by the methods outlined, his own average centring displace- 

 ment, r, and sighting-and-reading error, v, the surveyor can make use of relation (7) to 

 ascertain whether he is justified in neglecting errors of these magnitudes in the case of 

 any traverse angle he may select. 



The relative effects of v and r are best shown by means of curves, and this process 

 of analysis is perhaps most clearly elucidated by plotting curves for specific values of 

 v and r. In selecting three-sixteenths of an inch, or 0'016 foot, as a suitable average 

 amount for r, the writer believes himself safe from the accusation of having assumed 

 a degree of precision in centring beyond that secured in ordinary practice ; while 1 2 

 seconds probably approximates closely to a general average value of v for traversing, 

 when a reliable 5-inch theodolite is used, and when the angle is obtained by taking the 

 mean of single face-left and face-right observations. 



The curves of fig. 3 are constructed from relation (7), making use of these values of 

 v and r, and taking the lines enclosing the traverse angles as being of equal length, 

 i.e. L l =L. 2 . Curve A is plotted for a traverse angle of 180°, B for one of 120°, and 

 C for one of 60°, while the straight line D is for a traverse angle of 0°. 



These graphs, besides illustrating the decrease in importance of centring displace- 

 ments as the traverse angle becomes smaller, also show that while the centring error 

 has a preponderant influence when the lines are short, its effect rapidly diminishes as 

 they become longer. 



The necessity of close centring on short lines is generally admitted, and in 

 emphasising this fact, especially for traverse angles approaching 180°, lies the chief 

 practical value of these curves. They illustrate, for example, that with a traverse 

 angle of about 180°, and lines of less than 100 feet, nothing will be gained in accuracy 

 by an endeavour to reduce the average sighting-and-reading error, so long as the. 



