860 MR HENRY BRIGGS ON 



Now, this average displacement will be passed on to the next line of the traverse, 

 and, indeed, to all subsequent lines ; hence the end-point of the traverse will be affected 

 by an average error compounded of all such errors as v klL m + U^ii 2 . Thus, by ( 1 ) — 



Average total error at the \ /( ,, /T , T . t \ . 2/1-2 . T9 , t2\i /ir\ 



end of the compass traverse) = ±JiW+ + l+ ■ • • ■ + L») + « 2 (L? + 14 +U)} . (15) 



If the total length of the traverse, namely (Z 2 + L 2 . . . . + L n ), is constant, a choice 

 can often be made between performing the work by a few long lines or by a greater 

 number of shorter ones. Now, the magnitude of (L\ + L\ . . . . + LI) diminishes as the 

 number of lines increases ; therefore, in simple compass traversing, numerous short lines 

 are preferable to few long ones when the average total error is the question of chief 

 importance. This fact is pretty well known, but the graphical proof sometimes 

 attempted by writers, though plausible at first sight, is unsound and incapable of 

 withstanding careful scrutiny. 



(b) In the Case of a Theodolite Traverse. — By means of relation (12) the average 

 errors in the bearings of the traversedines are successively ascertainable ; similarly, the 

 average errors affecting the lengths may be determined from (13), writing k 2 in place of 

 k, where k 2 is the value the coefficient assumes for a steel tape. Using the same 

 notation as before, we have — 



Average total error at the end { //7 ., /T T , - (J 2ffi T . 2R , 2 a.t2o'*\\ /ifi\ 



of the theodolite traverse J " ± VW^i + I^ • • • • + K) + (W2 + ¥s • • • • +W-)) ( ib ) 



The total error may be analysed in the following way : — 



The co-ordinates of the end point of the traverse, with reference to the first point as 



origin, are — 



Latitude = Lj cos /3 1 + L 2 cos j3 2 . . . . + L„ cos /3„ ; and 

 Departure = Lj sin fi 1 + L L , sin /3 2 .... + L„ sin /?„. 



Therefore, by applying the theory of errors, we obtain — 



A i-erage error in the latitude of the end, point. 



= ±V{/^,cos' 2 /? 1 + L 2 cos 2 #, +L„cos 2 /3 ; ,) + (L:i i 8|sin 2 i 8, + L^3 2 sin 2 y83 +L£/S?aiii , fl,)} (17) 



A verage error in the departure of the end point 



= ±J{%(U sin 2 ft + L 2 siu 2 p 2 .... + L„ sin 2 p„) + (L 2 /3 2 2 cos 2 & .... +L 2 # 2 cos 2 /3„)} . (18) 



It is seen that (16) may be obtained by compounding (17) and (18). 



Of these last three equations (16) is the simplest and most useful ; it allows of an 

 assessment being made of the final error which might reasonably be expected in any 

 theodolite traverse, whether "closed" or "open.'' Relations (17) and (18) are of value 

 when attention is directed to the error in some particular direction, for the co-ordinate 

 axes can purposely be arranged so that the direction in question is either north -and- 

 south, or east-and-west, with reference to them. 



Il follows from (16), since /3' 2 , /3' 3 . . . . f$' H all attain their maximum values when 

 J\, To, 7' : > .... 7',, are all 180", that, other things being equal, a straight traverse will 



