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ADJUSTMENT OF RHOMBOIDS, ACKERMANN 79 
One could try to make this extremely simple solution suitable to practical application 
by introducing additional corrections. 
However, in this paper the adjustment of indirect observations is abandoned, and 
the further considerations go along other lines. 
3. Adjustment of conditioned observations. 
The second possibility for a least squares solution is the application of the adjust- 
ment of conditioned observations (or Standard Problem I according to Tienstra). It is 
the normally applied method of rhomboid adjustment and is based on the one condition 
implied in the ten directional observations. As all further considerations in this paper 
refer essentially to this method the general formulae are given. 
The condition involved results from the fact that by repeated application of the sine 
law the distance BE — e can be computed twice, via wing-point C and via wing-point D. 
Both computations should yield the same result, giving rise to the condition equation 
sin (2— 1) sin (10 — 9 4- 6 — 5) sin (3 —2) sin (7 —6 - 9—8) 
ECT d= SRI T) sin (9— 8) (3.1)* 
This equation can be brought to a more convenient form, for the subsequent lineari- 
sation, by expressing it logarithmically. Anticipating section 4 the natural logarithm is 
chosen here. 
In sin (2— 1) t- Insin (10 —9 + 6— 5) + In sin (9 — 8) + In sin (3—2+ 4 —T) 
—In sin (2—1 + 5—4) — In sin (10 — 9) — In sin (3 — 2) —In sin (7—6 + 9—8) = 0 (3-2) 
The substitution of the observed values of the directions 1,..., 10 into this equation 
will give the “logarithmic discrepancy” t: 
t = —In sin (2 —1) —In sin (10—9 + 6 —5) — In sin (9— 8) —In sin (3 — 2+4 — 7) 
tInsin (2—1 + 5—4) + Insin (10—9) + In sin (3—2) + In sin (7—6 + 9—8) 
(3.3) 
On introducing this logarithmic discrepancy t and the necessary corrections Viste» 
v, to the observations, the condition equation (3.2) can be linearized. Applying dif- 
ferentiation instead of using logarithmic table differences as is normally done, the linear 
etg (7—6 - 9—8) — etg (9—8) ]vg 
—etg (7—6--9—8) t etg (9—8) — ctg (10—9 -- 6— 5) -- etg (10—9) ]v, 
ctg (10 —9 + 6 — 5) — ctg (10 — 9) }v109 —t=0 
The linearized condition equation, written as in (3.4) or in its equivalent form with 
logarithmic table differences, will provide a rigorous least squares solution for the 
adjustment of rhomboids. Its performance will give corrected directions with which, 
finally, the coordinates of the points C, D, and E can be derived, making use of the 
points A and B which are supposed to be known from the preceding rhomboid. However, 
either form of the linearized condition equation still contains the logarithmic t. There- 
fore, if any attempt to avoid the use of logarithmic tables is to be successful, it will be 
necessary to bring it into another form which will allow other means of computation. 
form is: 
[ etg 2—1+4+5—4) —ctg(2—1)}v, 
Ff—etg (2—1+5—4) + ctg(2—1) — ctg(83—2+4—T17) + etg (3—2)}v, 
FÍ etg (3—2- 4—"7) — etg (3—2) ]v, 
{ ctg (3—2- 4—7) * etg (2—1-- 5—4) qv, 
Ff—etg (10—9 + 6—5) — ctg (2—1 +5—4) }v, ; 
Ff etg (10—9 + 6—5) + etg (T—6 + 9—8) jv, (3.4) 
+{—ctg (3—2 +4—7) — ctg (7T—6 + 9—8) }v, 
H 
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* See: Reicheneder, Fehlertheorie und Ausgleichung von Rautenketten, Berlin 1949.