AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 857 



If this criterion of negligibility be first applied to determine for what length of 

 traverse line, L, a sighting-and-reading error of 12 seconds (5"8 x 10~ 5 rads.) may 

 be neglected when r = 0'016 foot, and T= 180°, we have : 



v< \{^}> 



giving L < 114 feet as the result we require. A further justification, or, properly, 



an excellent excuse, may here be deduced for taking v as constant for all values 



of L, or, in other words, for drawing D, fig. 3, as a straight line ; for though v, 



strictly speaking, must increase to some extent as L becomes shorter than 100 feet, 



it cannot do so at the same rate that the curve shows the centring error to increase. 



Even if v were to become 60 seconds for traverse lines 24 feet long, or five times the 



amount selected in constructing the curves, it would nevertheless be almost negligible 



under the above criterion when r=0"016 foot and T= 180°. It would therefore 



be of no immediate consequence to determine — were it possible — the rate of increase 



of v as the lines become very short, since the increase can have very little, if any, 



sensible effect. 



The same criterion also allows of a test being applied on the legitimacy of the 



assumption, already made, to the effect that triangulation lines are usually of such a 



length that a small centring displacement has no tangible influence. An average 



12 

 sighting-and-reading error of 12 seconds will be reduced to y=, or 7 seconds, by three 



reiterations. Consider the case of two sides of a triangle of equal length, L, embracing 

 an angle of about 60°. If r=0016 foot, L will be such that the average angular 

 centring error is equal to or less than one-third of 7 seconds (expressed in radians) when 

 L > about 900 feet. As the sides of the main triangles in a minor triangulation 

 scheme generally exceed 300 yards in length, and, moreover, since it is usual — 

 though evidently not strictly necessary — to centre with greater care in triangulation 

 than in traversing, we may safely conclude that centring errors in triangulation are 

 negligible in their effects when compared with those we have termed sighting-and- 

 reading errors. 



Therefore we are justified in assuming, as we shall do when discussing the influence 

 of errors in triangulation, that the average error in each of the angles of a triangle is of 

 the same magnitude, being, in point of fact, equal to v, the sighting-and-reading error. 



Section IV. The Propagation of Errors in Traversing. 



Two classes of errors have to be taken into account in determining the average error 

 at the end of a traverse consisting of several lines — namely, errors in bearing, and 

 errors in linear measurement. Before it is possible to deduce expressions compounding 

 these errors, it is first necessary to consider them separately : — 



TRANS. ROY. SOC. ED1N., VOL. XLVII. PART IV. (NO. 28). 126 



