836 Mr Sang on Optimum Surveying. 



tions A, B, C, &c., have been absolutely determined ; but these 

 positions also may be liable to uncertainty. The whole sys- 

 tem of stations are connected by a series of traverse lines ; and 

 the question of optimum determination may be extended thus ; 

 to find those positions of a multitude of stations which may 

 best agree with the observed angles. 



To suit the above investigation to this inquiry, we have only 

 to denote hy ah the probability of error in aim, as known from 

 the field-book, on the bearing AB. We shall then have 

 ^ ^ sin BA 2 . ^ sin BA . cos BA , 



^'^''^ -^6^ ('''^-^b) =2 __ (,^_ y^) 



. 2 sin BA . cos BA , ^ -, cosBA^ 



^ ^ sin AB- , - X. sin AB . cos AB . 



^ ^J sin AB . cos AB , ^ cos AB^ 



^^"^ " ^p ("" ~\= — ^t-cye-yA) 



etc. etc. 



Here, then, are two equations for each station, and it might 

 therefore appear that the positions of all the stations may be 

 determined from the bearings alone ; but it must be observed 

 that the entire sum of the equations depending on J ^ is zero, 

 as well as of those depending on dy^ so that two of the num- 

 ber need not be counted : and again, that, as there is no abso- 

 lute term, only the ratios x — x :y — y -.x — x -, ii — y ^ 



^ J A B -^A "^B A C »>'a "^C' 



&c., could be got. Thus we are at liberty to assume co-ordi- 

 nates at will for any one station, and to introduce one arbitrary 

 condition ; which arbitrary condition might be the length of 

 the base line, or the distance, any how determined, direct, in 

 latitude or in longitude, between two stations. 



The capabilities of this method include the best possible so- 

 lution of every case that can occur in plane surveying. 



If, for example, the latitudes of some stations, the longitudes 

 of otliers, and both latitudes and longitudes of a third class, had 

 been determined by other processes susceptible of a certain 

 degree of precision, these determinations could be combined 

 with the bearings. Let X be the so-determined latitude of 

 A J and a the chance of error on it, the term 



*A — X 



