B 
EC TE 
Referring also to Figure (1), one gets 
Zi = Y: cosec O0, tan y; (5) 
Zi-V = Y: cosec a» tan ys (6) 
From equations (5) and (6), one gets 
2, = 0.5 Y: (cosec à; tan y; + cosec da tan Y,) + 0.5V 
Substituting the value of Yi: from equation (4) into the above equation, one 
gets 
nn. 0.5B 
bd o — + + ; 
i Cot ài - cot Oz (cosec a; tan y; + cosec oy tan yy) + 0.5V (7) 
  
Equations (5), (6) and (7) give the coordinates of point i. 
III. THE STANDARD DEVIATIONS OF TARGET COORDINATES OX, 9r, AND 
02. 
i 
Applying Taylor expansion to equations (3), (4) and (7), one gets the 
errors VXi, VYi and VZj as functions of the errors of theodolite directions 
Va, Yao; Vy: and YY» 
  
  
  
-B 2 
. = + 
VX, (cot o, 4 Cof az)? (cot as cose 0; Vo cot a; cosec® ap Vas) 
(8) 
VY. = B (cosec? a, Vo, - cosec? a; Va») (9) 
i (cot a; - cot ay) 
yz, = : 2 [(cot Q1 - Cot Q5 ) 0.5B 
i (cot.04. - cot 0) 
(-tan y; cosec a; cot a; Vo; - tan Y2 cosec dy cot 0, Vo, 
+ cosec 0; sec?y; Vy; + cosec À, sec! y.Vy,) - 0.5B (cosec y, 
2 
tany; + cosec à», tan yo) (-cosec?^o, Vo, + cosec®ay Vo) 
The above equation VZ. can be reduced to the following form : 
i 
(2: - 0.5V) 
  
: 2 E - 
Va, = [= 0.52; cot o *6et ur o az) Sosec a1] va, + [ 0.5(Z, -V) 
(Z2. = 0.5V) 2 
> i 2 Sec Yi 
-— ee ‚52. — — 1 V 
cot a, GET IST 85 fosse 02] Vo, + [0.5 i te zi] v1 
2 
sec 
+ [0.5(z, - V) Fe] Vy, (10) 
The standard deviations ox, oY, and oz; obtained from equations (8), (9) and 
(10) take these forms 
p? 
2 2 2 2.2 
. = Cot à» cosec^ ài * cot a; cosec^ a2)^o*a (11) 
G x (cot «1 - cot 02)” ( 2 } 2 ) 
  
B? 
2 4 4 2 
a = cosec' d] * cosec* o»2)o^o 12 
o (cot a4 - cot o5)" ( I 2) (12) 
  
Subst: 
equati 
Zi, Ol 
where 
The al