AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 859 



Then the bearing fi mt of any line, is given by — 



p m = ft + T, + T, . . . . + T m _j - 180°(m - 1 ), and 



ft,= ± j{t\+ti .... +ti_ x } (9) 



From (7) we have that the average error in any angle — 



/ f , 4rV 1 1 2 cos T m ) 



Combining expressions (9) and (10) — 



/ I . ,. , 4rV 12 2 2 1 \ 8?- 2 /cos T, cos T, cos T„, A ) 



This may also be written — 



The latter form is of more service in studying in detail the propagation of angular error 

 in an actual traverse, since the average error of the bearings are then worked out in 

 succession, commencing with /3' 2 . 



By means of equations (11) or (12) the average error in summation of the angles of 

 a closed traverse, or polygon, can be determined. 



(2) Errors in Linear Measurement. 



If a line were 660 feet long it would require a 66-foot chain or steel-band to be 

 applied ten times to cover it ; hence if the error of a single application of the chain or 

 band were a, that for the whole line would be 'a >J\Q. Therefore, if L be the length of 

 a line, and / the average error in that distance. 



Z= ±ksjL ...... (13) 



The coefficient, h, is often taken as a constant for a given kind of instrument (chain or 

 tape), but, strictly speaking, its value is dependent also on the length of the chain or 

 band used, being smaller for a long than for a short one. 



Average Total Error at the End of the nth Line of a Traverse. 



(a) In the Case of a Simple Compass Traverse. — Consider the case of a traverse 

 in which bearings are taken by a compass and lengths by an ordinary chain. Let u be 

 the average error in each bearing, and k^ the value k assumes for a chain. Let L m be 

 any line of the traverse, and l m the average error in its length; then, from (13) we 

 have — 



/,„=±Wt (14) 



Owing to the error in bearing, the end of any line L m will be displaced by the average 

 amount L m u. Therefore the displacement of the point will be due to the resultant of 

 the vector errors l m and L m u : that is to say, it will be zbv^fL TO + L^w 2 , by equation (1), 



