Adams. — On Triangulation by Least Squares. 199 



pendent of the observed angles, as the same observed angles 

 have been used in each case. Each adjustment gives a 

 consistent geometrical figure, and preference is given to 

 the least-square adjustment because it gives this consist- 

 ent figure with the least alteration to the observed angles, 

 as shown in this example, and as would also appear by 

 comparing any other process of adjustment therewith. 



The notation used in the least-square adjustment is similar 

 to that used in example No. 1, and is here repeated for con- 

 venience of reference. 



Let I 1 = length of P P 4 calculated from P P 1; using the 

 angles from (6) Schedule 1, (5) Schedule 2 ; 

 „ i = true length of P P 4 : 



I — I 1 

 then e = —: radians 



I 

 € — sum of angles at P (from (4) ) — 360° 

 ai = cot Aj 

 /Si = cot Bj 

 a x = 2aj + /3 X 

 h = —ay- 2ft 

 Ci = — ai + ft 

 2k = %% (a 2 + b' 2 + c l ) 

 h — Ci + c 2 + c 3 

 i = the number of triangles. 

 The equations for P and Q are — 



//P+ 2iQ+e =0) 

 2 k P + h Q + e = J 

 The corrections to the angles are — 



x\ = a,P- Q 



yi==6iP-Q 



^ == c,P +2Q, &c, 



and are given in (6) ; and the corrected angles are — 



A x + a?! 

 B x + //i 



d + «!, &C, 



where the corrections are applied to the values of the angles 

 in (5). 



For the theory of the adjustment reference must be made 

 to any of the treatises on least squares. The method here 

 used is described by Colonel Clarke in his " Geodesy," but 

 differs in the application of the triangular error, which is 

 applied before the condition - equations are derived. This 

 shortens the numerical work considerably, and thereby lessens 

 the risk of numerical slips. As in the case of example No. 1, 

 all the calculations have been performed on the Brunsviga 

 calculating-machine. 



