342 Mr Sang on Optimum Surveying. 



plane surveys. Dividing it by the probability (A N) that the 

 traverse will pass aside of the station, and summing the squares 

 of these quotients taken for every observed bearing, we have 

 the general measure of inaccuracy ; this is to be a minimum. 

 The quantities here to be determined are the latitudes and 

 longitudes of the stations, and the radii a, j8 of the spheriod. 

 Differentiating, then, according to these unknown quantities, 

 and separating the terms containing the independent variations ; 

 observing also, that we are at liberty to take either a or jS as 

 known, or as unit, we obtain as the most exact determination 

 of the various quantities sought, the equations — 



sin A N 



\ sin lat N^ cos lat N \ sin lat A . cos (Ion A — ^lon N) 



\sti Depending on b . (3-. 



. /n » T ^TN cos A N ) , . , , ^.^ T X TVT9 1 i. A sin A N ) 

 + sm (Ion A — Ion N) ^ — J- + sin lat N . cos lat N^ cos lat A . — - — v 



i /32 . , ^ sin A N 

 -f 2 . ,^r=^ Sin lat N^ . cos lat A . 5 — 



a2_i («2_^2) sin lat A^ . , ^ . . ^ ^ sin A N 



-4- 2 . ^-^ — jr^^r^ Sin lat A . cos lat A . s — 



' (A)3 e^ 



= 0. 



where the e denotes the chance of error on the bearing A N, 

 that is the probable distance at which the traverse line will 

 pass aside of the station. 



^d, Depending on longitude of place of observation ; d Ion A. 

 (1*^ part) 



^2 f sin A N 

 2 . T^ < cos lat N . sin lat A , sin (Ion A — Ion N) . — 



COS A. N^ \ 

 H- cos lat N . cos (Ion A — Ion N) — r = ^ 



3^, Depending on longitude station observed ; 5 . Ion N. 



a^ f 



2 . Tj^x S — ■ COS lat N . sin lat A sin (Ion A — Ion N) 



— cos lat N . cos (Ion A — Ion N) '^^^-^ — r = O 



e' ) 



In these equations (the 2d and Sd) it is to be observed, that 

 the sign S applies only to part of the survey, and that d Ion A 



sin A N 



