Adams. — On Triangulation by Least Squares. 197 



applied to (11) to obtain the sines of the final angles shown 

 in (13). 



The corrections to the sines in (11) are shown in (12), and 

 are equal to (7) x (10). 



The triangles are now solved, using the final angles in (8), 

 and the results are given in (15). 



The bearings in (16) are obtained by applying the final 

 angles in (8) to the given bearing of P P 1( checking on the 

 bearing of P P 4 . 



The rectangular co-ordinates of P., and P R are calculated 

 from the bearings and distances in (16) and (15). 



This completes the adjustment of the polygon. It will be 

 seen that all the geometrical conditions of the figure are com- 

 pletely satisfied, but, as will be shown subsequently, the 

 corrections applied to the observed angles are considerably 

 larger than those required by least squares. It will also be 

 noticed that the centre angles are corrected independently of 

 the side adjustment, which is thus not allowed to influence 

 the adjustment of the centre angles in any way. 



II. The Least-square Adjustment. 



The application of this adjustment is given on the schedule. 

 Columns (1), (2), (3), and (4) are the same as in the ordinary 

 adjustment. The angles of the first computation in (5) are 

 not corrected for the error of the centre angles, but are equal 

 to (2) + (3). (9) contains the sines of (5), and with these 

 sines the three triangles are solved, and the length of P P 4 

 obtained by calculation from P Pi. 



Comparison of this value with the true value gives t, while 

 e is obtained from (4). 



Column (10) gives a u b u c 1; &c, and 2 (a 2 + b 2 -f c 2 ) ; from 

 this column h = c x + c 2 + c 3 and 2 k = -J 2 (a 2 + b~ + c' 2 ) are 

 obtained. 



The equations for P and Q are now formed and solved. 

 With these values of P and Q the corrections to the observed 

 angles are calculated, thus : — 



and the values are entered in column (6) and applied to the 

 angles in (5), giving the final angles as shown in (7). 



A check is obtained at this stage by noting that P : P P 4 + 

 C x + C 2 + C s (from (7) ) = 360°. (11) contains the sines of 

 the angles in (7), and a check computation of P P 4 from P P 1 

 proves the correctness of the work. 



The triangles are solved using the sines in (11), and the 

 results given in (13). (14) contains the bearings obtained by 



