176 GEODESIC INVESTIGATION'S. 



With first-class instruments, find by means of actual observation fusing the 

 method of lea^t squares, &c.) the latitudes of the stations, and their three 

 pairs of azimuths. (It is supposed that the observer knows how to use the 

 instrunients, so as to obtain correct readings of angles of altitude and azimuth, 

 no matter what may be the deviations of the plumb-line from the normals at 

 the stations). 



Let V, I" be the latitudes of the stations S^, S-i. 



Let A' be the azimuth of S% as taken at S-^ ; and let A" be the azimuth of 

 S\ as taken at S.2, . 



Then we know the values of the four entities — 



I', I", A', A". 



. Assume each tliree of these four entities as data, and compute therefrom 

 the obtainable volues of the latitudes and azimuths, which (written in order) 

 vre may denote by— - 



^ i> ^ 1) ^ 1) ^ 1- 



Find also the four obtainable values of the difference of longitude of the 

 stations. 



From the observed and computed values of the latitudes and azimuths, and 

 the computed values of the difference of longitude of the stations, find the 

 most probably correct values ; and represent them in order by — 



? 2) ^ 2) -4 2) ^ 2) '•'2' 



Assume each three of these five entities as new data, and compute the/owr 

 values of each entity obtainable therefrom ; and find the most probable value of 

 each entity from aKthe known values (observed and computed). 



Represent the values so derived by — 



^3) ' 3) ^ 3) ^ 3j (^3- 



Assuming these last as more compatible values of the latitudes, azimuths, and 

 difference of longitude, proceed in like manner, and find a new set of values 

 for the fire entities. And represent such values by— 



^ 4) ^ i) ^ 1) ^ 4) '^f 



Assuming these last as still more compatible values of the latitudes, 

 azimuths, and difference of longitude, proceed in like manner, and find a new 

 set of values for the five entities. 



Represent such values by — 



' fi) ' 5) ^ 5> -^ a) ^5- 



Now, under the circumstances (in which the stations are supposed to be 

 chosen by a competent person, and the observations taken with the greatest 

 care), it will be found that this last set of values of the five entities are equals 

 respectively to the immediately preceding set, or so nearly so as not to differ 

 from them by amounts appreciable in the most rigorous practice. 



Omitting the subscript figures, we may write the last obtained values of the 

 latitudes, azimuths, and difference of longitudes, in the adopted notation, as 

 follows : — 



' 1) ' 1) ^ 1) -^ 1) '^i.- 



