Mr Sang on Optimum Surveying. 339 



from which three equations, the co-ordinates x^^ y^^ and the ar- 

 bitrary y, which is the angle between the assumed and the true 

 meridian, can be determined. 



It often happens that merely angles are observed at a station. 

 That is to say, an assumed direction is taken for the meridian ; 

 or the bearings are taken by the help of a bapk observation. 

 Hence, there enter other questions than those which I have 

 solved in p. 24, for at each station A, there is an uncertainty a 

 in the bearings. Allowing for this, the formulae will stand 

 thus : — 



oei' i S _ i ^sm(AB— «).cos(AB — «) 



~ ^6^ {cos(AB — a)2--sin(AB — «)«} 



a^A;2.^^2— sin(BA — /S)2=2.^^-^sin(BA — S).cos(BA — A) 

 a2/A;2.!iII^sin(BA — /3).cos(BA-^)=2.?^-?^cos(BA-/Jl2 



In these expressions, when the bearings taken at a given sta- 

 tion have been obtained from a direct astronomical observation, 

 there is no arbitrary correction a to be applied : but when the 

 position of the meridian has been assumed merely, or has been 

 determined by transference from some remote station, the cor- 

 rection a ought to be used. In the case of pure assumption, a 

 must be treated as a large angle ; but in the case of transfer- 

 ence, it may be treated as a small one ; in which case, it will be 

 sufficient to regard its cosine as unit, and its sine as equal to 

 the arc. 



The above equations used in this way will include the 

 best possible determination of the positions of the stations, 

 taking all the circumstances of the angular observations into 

 account. 



I have now described the method of Optimum Surveying as 

 applicable to plane surveys ; before proceeding to inquire 

 whether it be capable of extension to geodetical operations, it 

 may be worth while to glance at the advantages which it ofFei-s, 

 that we may be fortified against any formidable difficulties. 



A net- work of traverse lines has been thrown over the whole 

 extent of the British Islands ; the bearings of many of these 



