6 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 66 



usually attendant on such observations are not properly differences 

 of azimuth, but small spherical arcs which correspond to very dif- 

 ferent azimuths according to the zenith distance. 



From the equations ( i ) , allowing different weights to the individual 

 observations according to the method of least squares, we get ALo 

 and A^o and therefore also Lc and ^e with their mean errors. 



The final azimuths are, therefore, 



Aie^A^ie + Vi sec hie. 



In case the vi's are too large it is to be recommended that Aie should 

 be computed again directly from the final Le and ^e. 



Of course we determine x and y in the units of the absolute term 

 Oi. In this respect we can introduce various modifications according 

 to circumstances. 



If we designate by in the spherical distance of the approximate 

 terminal point Eo{Lo, (j>o) from the observed direction designated by 

 A'ie, then we have approximately o>i = fii cosec u. 



If we then put 



cos, hie ^r 1 



sinY = ^^"^ _ (3) 



COS ^°ei = ai, sin ^°ei=—&i J 



we get ^ 



z'i = Vyt(/ti + ai.r + &iy), (4) 



and Vyi is a weight factor for the whole equation. 



If we desire, for example, when reading from the chart for ju,i, as 

 our unit a measure of length which shall then also hold good for x and 

 y while the v's shall be given in degrees, so that in general the units 

 may be taken to be one kilometer for altitude and one degree for 

 azimuth, then the above-giVen equation for Vyi will still have to be 

 multiplied by the following factor 



— — — =0.0089807, 

 III-35 



where the earth's radius is assumed to be i? = 6,380 km. 



It is easy to see that the coefficients ai and hi can, therefore, with 

 advantage, be read off graphically, especially when the observations 

 do not deserve the most accurate computational methods. 



If we also find it advisable to use the parallax in altitude in the 

 determination of the location of the terminal point (Le, ^e), which 

 as we have already mentioned, is scarcely avoidable, then the approxi- 

 mate value Ho must be deduced more accurately and the consequent 

 apparent altitude hie corresponding to the preliminary distance Die 

 must be computed more carefully. 



