338 Mr Sang on Optimum Surveying. 



When three signals are observed, only one result can be ob- 

 tained, but when four or more are noticed, the combination of 

 these three and three would give so many positions. If, in fact, 

 upon two signals as extremities, an arc be constituted, contain- 

 ing the measured angle, that arc will be one locus of the sta- 

 tion sought. In this way, by taking each pair of signals, we 

 obtain a new locus. These loci will not, probably, all meet in 

 one point, and the true position of the station must be sought 

 in a manner analogous to that already given. 



The observations at the station N do not give the bearings 

 of the signals, but only the differences between the bearings : 

 hence, assuming the direction of an arbitrary line for the meri- 

 dian, the observed bearings N A, N B, &c., will differ from the 

 true bearings by an angle v peculiar to the station N ; so that 

 the true bearings are (NA — i'), (N B — v), &c. ; and here v is 

 not to be taken as a small correction ; it may be an angle of any 

 magnitude. 



These bearings, referred to the other extremities of the lines, 

 become augmented by 180°, or the signs of their sin and cos 

 changed : hence, the perpendiculars let fall from N upon paral- 

 lels down through A, B, &c. are of the form 



— (-^"n — -^a) sin (N A — v) + {y^ — y) cos (N A^ — v) 

 and thus we must have for a minimum the sum 



_N A_^j4 isin(NA — v)cos(NA — v) I 



an an ^ '^ / 



+ 2 . (""iZlt . cos (N A — v) ^ 

 \ an ' / 



where p also is a quantity to be determined : we should thus 

 have the equations — 



= 2 2. J? .''^ licos2(NA-v) 



an an 



Ja:j,;5.!!iZp^.sin(NA — v)2 = 2.^-^-.sm(NA— v).cos(NA — v) 

 a » « » 



Syj,;2.!!Ll^sin(NA — v).cos(NA — v) = 2.5^— ^^-^cos(NA — 0' 



J 



