AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 861 



be affected with a greater average total error than any other traverse with the same 

 number and lengths of lines. 



The relative Accuracy of Compass and Theodolite Traversing. 



Consider a straight traverse of n equal lines, running between two fixed points W 

 feet apart. In these circumstances nL= W (L being the length of any of the lines), 

 and T 7 ! = T 2 . . . . = T n _ x = 180°. Therefore (15) reduces to— 



Average total error in a simple compass traverse = ± ^{JjW + w 2 LW} . . (19) 



In like manner (16) reduces to — 



Average total error in a theodolite traverse^ ± I < JczW + — i 1 -\v 2 + -^-}\ \ . (20) 



Now, while (19) diminishes as L diminishes, (20) increases as L diminishes; thus 

 keeping W constant, it follows that there must be some value of L for which the average 

 total error of the theodolite traverse equals the average total error of the compass 

 traverse, and this condition will be secured when — 



In these circumstances L will be short, and unless the fixed points between which 

 the traverses run are very close together, it is sufficiently exact to assume (W— L) as 

 equal to W; when the above simplifies to the following cubic equation : — 



L 3 +L2 | 2(^-^)-W > 8W = Q .... (21) 

 ( 2w 2 J iru 



To illustrate the value of this result, let us make a comparison between the miner's 

 dial (used as a simple compass instrument) and a 5 -inch theodolite, when our experience 

 with the instruments has led us to conclude that k x (for the 100-foot chain) = 0*0096, k 2 

 (for the 100-foot steel band) = 0*0063, v=12 seconds, and u = \°. Instead of taking 

 v = ^q inch, as has been done in former examples, we may profit by the results attained 

 in an earlier part of the paper, and, now that we are dealing with exceptionally short 

 lines, reduce the average centring displacement to ^ inch. Substituting these values in 

 (21), and taking W, the total length of the traverse as 1000 feet, L is found to be 

 approximately 23 feet. In other words, under the stated conditions, if the lines of a 

 straight traverse, all assumed equal, are more than 23 feet long, the theodolite may be 

 expected to give more accurate results than the miner's dial ; yet if they should be 

 shorter than this figure the compass instrument will have the advantage, providing local 

 magnetic attraction is absent. It also follows from (21) that this limiting length of line 

 increases as W increases ; that is to say, the relative accuracy of the two modes of 

 traversing depends on the total length of the survey, being more in favour of the com- 

 pass method with long traverses than with short ones. 



Before leaving this section of the subject the writer desires to make it quite clear 



