196 Transaction*. — Miscellaneous . 



scribed, and then the least-square adjustment will be con- 

 sidered and the practical application of it explained. 



In the figure the sides P P x and P P 4 are derived from the 

 existing triangulation, and are to be adopted as correct both 

 in bearing and distance. The triangulation is to be extended 

 to include the stations P 2 and P 3 , and to this end all the 

 angles of the triangles are equally well observed. These 

 observed angles are shown in column 2 of the schedule. 



I. The Ordinary Adjustment. 



The first correction to the observed angles consists in 

 applying one-third of the triangular error in each triangle to 

 each angle, and is shown in (3). The corrected centre angles 

 are entered in (4) and added to the given angle Pj P P 4 , and 

 the sum of these angles should be 360°. This is not usually 

 the case, so the difference between the sum and 360° is dis- 

 tributed equally among the centre angles : thus each centre 



e 







angle receives a further correction of - where c = the dif- 



ference between the sum of the angles in (4) and 360°, and 

 i = the number of triangles. To keep the sum of the angles 

 of each triangle equal to 180° it is necessary to apply half 

 this correction to each of the base angles. These corrections 

 are shown in (5), and the corrected angles appear in (6). 

 This completes the adjustment of the centre angles, and the 

 subsequent corrections affect the base angles only. 



The length of the side P P 4 is calculated from P Pj using 

 the angles in (6) — the sines of these angles appear in (11) — and 

 the length so obtained is compared with the true length. As 

 the two values of P P 4 do not usually agree, a further cor- 

 rection to the base angles becomes necessary, and is found 

 thus : If / = true length of P P 4 and I 1 = length of P P 4 



I - I 1 

 calculated from P P 1; using the angles in (6), then e = — -, 



radians. In (9) the cotangents of the base angles are given 

 and the sum 2 (cot A + cot B) obtained. Then the correction 



to each base angle is = =-7 r— 1 u , radians. 



i (cot A + cot jB) 



The calculation of the correction is shown on the schedule, 

 and gives 4"06 in this example. In (7) the correction is 

 applied, and the final angles appear in (8). 



At this stage the work is checked by calculating P P 4 from 

 P P,, using the final angles in (8), and. as shown on the sche- 

 dule, the calculated value of P P 4 agrees with the true value, 

 thus proving the correctness of the work. 



In (10) the value of 1" for each of the base angles is given. 

 The products of (7) and (10) give in (12) the corrections to be 



