75 28 42 83 � AOBweight 3 26 43 15 102� BOC weight 2 22 27 38 94� CODweight 4

75 28 42 83 ? aobweight 3 26 43 15 102? boc weight 2

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75 ". 28 ' 42 83 ο = AOB weight 3 26 ". 43 ' 15 102 ο = BOC weight 2 22 ". 27 ' 38 94 ο = COD weight 4 77 ". 23 ' 23 79 ο = DOA weight 2 Adjust the angles by method of Correlates. Solution: 75 ". 28 ' 42 83 ο = AOB Weight 3 26 ". 43 ' 15 102 ο = BOC Weight 2 22 ". 27 ' 38 94 ο = COD Weight 4 77 ". 23 ' 23 79 ο = DOA Weight 2 ____________________________________ Sum = 00 ." 03 ' 00 360 ο Hence, the total correction E = 360º - (360º0’3”) = -3” Let e 1 ,e 2, e 3 and e 4 be the individual corrections to the four angles respectively. Then by the condition equation, we get e 1 + e 2 +e 3 + e 4 = -3” -------- (1) Also, from the least square principle, Σ(we 2 ) = a minimum 3e 1 2 + 2e 2 2 +4e 3 2 + 2e 4 2 = a minimum ------- (2) Differentiating (1) and (2), we get
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δe 1 + δe 2 + δe 3 + δe 4 = 0 --------- (3) 0 2 4 2 3 4 4 3 3 2 2 1 1 = + + + e e e e e e e e δ δ δ δ --------- (4) Multiplying equation (3) by –λ and adding it to (4), we get δe 1 (3e 1 – λ) + δe 2 (2e 2 -λ) + δe 3 (4e 3 -λ) + δe 4 (2e 4 -λ) = 0 --------- (5) Since the coefficients of δe 1 ,δe 2 ,δe 3 ,δe 4 must vanish independently, we have 3e 1 – λ = 0 or e 1 = 3 λ 2e 1 – λ = 0 or e 2 = 2 λ ---------- (6) `4e 1 – λ = 0 or e 3 = 4 λ 2e 1 – λ = 0 or e 4 = 2 λ Substituting these values in (1), we get 3 2 4 2 3 - = + + + λ λ λ λ 3 ) 12 19 ( - = λ 19 12 * 3 - = λ Hence " 63 . 0 19 12 19 12 * 3 * 3 1 1 - = - = = e " 95 . 0 19 18 19 12 * 3 * 2 1 2 - = - = = e " 47 . 0 19 9 19 12 * 3 * 4 1 3 - = - = = e " 95 . 0 19 18 19 12 * 3 * 2 1 4 - = - = = e _______________ Sum = -3.0” _______________ Hence the corrected angles
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AOB = 83º42’28”.75 – 0”.63 = 83º42’28”.12 BOC = 102º15’43”.26 – 0”.95 = 102º15’42”.31 COD = 94º38’27”.22 – 0”.47 = 94º38’26”.75 DOA = 79º23’23”.77 – 0”.95 = 79º23’22”.82 _______________ 360º00’00”.00 8. The following round of angles was observed from central station to the surrounding station of a triangulation survey. A = 93º43’22” weight 3 B = 74º32’39” weight 2 C = 101º13’44” weight 2 D = 90º29’50” weight 3 In addition, one angle ( ) B A + was measured separately as combined angle with a mean value of 168º16’06” (wt 2). Determine the most probable values of the angles A, B, C and D. Solution: A + B+C+D = 359º59’35”. Total correction E = 360º - (359º 59’ 35”) = + 25º Similarly, ( ) B A + = (A+B) Hence correction E’ = A + B - ( ) B A + = 168º16’01” – 168º16’06” = -5” Let e 1 ,e 2, e 3 ,e 4 and e 5 be the individual corrections to A, B, C, D and ( ) B A + respectively. Then by the condition equation, we get e 1 + e 2 +e 3 + e 4 = -25” -------- (1(a)) e 5 – e 1 – e 2 = -5” -------- (1(b)) Also, from the least square principle, Σ(we 2 ) = a minimum 3e 1 2 + 2e 2 2 +2e 3 2 + 3e 4 2 + 2e 5 2 = a minimum ------- (2) Differentiating (1a) (1b) and (2), we get δe 1 + δe 2 + δe 3 + δe 4 = 0 --------- (3a)
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δe 5 - δe 1 - δe 2 = 0 ---------- (3b) 0 2 3 2 2 3 5 5 4 4 3 3 2 2 1 1 = + + + + e e e e e e e e e e δ δ δ δ δ --------- (4) Multiplying equation (3a) by –λ 1 , (3b) by -λ 2 and adding it to (3), we get δe 1 (3e 1 –λ 1 2 ) +δe 2 (2e 2 1 + λ 2 ) + δe 3 (2e 3 1 ) + δe 4 (3e 4 1 ) +δe 5 (-λ 2 +2e 5 ) = 0 --------- (5) Since the coefficients of δe 1 ,δe 2 ,δe 3 ,δe 4 etc. must vanish independently, we have 0 3 1 2 1 = + + - e λ λ or 3 3 2 1 1 λ λ - = e 0 2 2 2 1 = + + - e λ λ or 2 2 2 1 2 λ λ - = e 0 2 3 2 = + - e λ or 2 1 3 λ = e ------------ (6) 0 3 4 1 = + - e λ or 3 1 4 λ = e 0 2 5 2 = + - e λ or 2 2 5 λ - = e Substituting these values of e 1 ,e 2, e 3 ,e 4 and e 5 in Equations (1a) and (1b) ) 1 ( 25 3 2 2 2 3 3 1 1 2 1 2 1 a from = + + - + - λ λ λ λ λ λ or 25 6 5 3 5 2 1 = - λ λ 5 6 1 3 2 1 = - λ λ -------- (I) ) 1 ( 5 32 2 3 3 2 2 1 2 1 2 b from - = + - + - λ λ λ λ λ 5 6 5 3 4 1 2 - = - λ λ --------- (II) Solving (I) and (II) simultaneously, we get 11 210 1 + = λ 11 90 2 + = λ Hence 64 ". 3 11 " 40 11 90 . 3 1 11 210 . 3 1 1 + = + = - = e
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` 45 ". 5 11 " 60 11 90 . 2 1 11 210 . 2 1 2 + = + = - = e 55 ". 9 11 " 105 11 210 . 2 1 3 + = + = = e 36 ". 6 11 " 70 11 210 .
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