yz. 
n= 
| two 
(4.1) 
(4.2) 
4.4) 
lace 
ms) 
the 
sing 
»dt 
tion 
end 
f a 
oef- 
sful 
xpe- 
stly 
Lion 
5.1) 
for 
— 
  
ADJUSTMENT OF RHOMBOIDS, ACKERMANN 81 
t 
Its solution, K = 64 directly yields the following corrections to the observations 
Uy = agio = : ud + 0 = + Vg = ++ 
32 2 82 32 9 16 
U 4 t V4 = — : mm t = : : eo 
2 16 4 82. "e à 5 Tn 
Realizing, at this stage, that with a slight extension of the computations necessary 
to get the linear discrepancy Ae, preliminary coordinates vor 
0 ? 
xp and £i of the points C, D, and E become known, it is obvious that the least squares 
D ) 
"Ep Ups Tp» Yy 
0 0 0 € € 
solution (5.8) can be carried through to give corrections to the preliminary coordinates 
directly. To achieve this, the formulae for the sine law and the coordinate computations 
have to be linearized. Assuming again, for the coefficients in these linear relations, those 
values valid for the ideal-shaped rhomboid, the resulting corrections (expressed in the 
coordinate system indicated in Fig. 1) will be 
C le d le 
de, = — . da — . des. = da, —0 
( e 4 D e À E E, 
5.4a 
c le : d le ( ) 
dy, = — m NY + . dy, — — 1 4e Y, = +4 
( e 16 D e 16 Vg PA Wr * à 4e 
Hence 
qc Bor F dr, xp =, t de; tn = 4 (x, + x, ) 
( 0 ) D 
= 1 (5.4b) 
Voy, + dyg yp y, t dy, Ya (y, ty, ) 
0 0 C D 
Summarized, the computations required for this simplified rhomboid adjustment are: 
a. Computation of the preliminary coordinates of the points C, D, and E using the 
observed directions. This gives as a by-product the linear discrepancy Ae. 
b. Correction of the preliminary coordinates according to (5.4), if necessary. 
In this method the main amount of work consists of computing the preliminary 
coordinates. The adjustment then, which itself is hidden, and does not show up any more 
as a separate step in the computation, yields in a very simple way the corrections to 
the preliminary coordinates. 
It should be noted that this simplified adjustment method approximates much more 
closely to the rigorous solution than the method mentioned and abandoned in section 2, 
because the preliminary coordinates, on which the adjustment is based, give a much 
better approximation to the actual shape of the rhomboid. From experience it can be 
taken that the linear discrepancies Ae are normally in the order of 10—3 to 10—4 parts 
of the base b. Therefore this adjustment method should cover most practical cases. It 
can be fully recommended, if the rhomboids are well shaped or alternatively where the 
accuracy of the solution is of less importance than a direct and simple solution for the 
coordinates. 
Remark. It should be noted that the size of the linear discrepancy Ade shows immediately 
if the adjustment is necessary at all. It serves as a criterion to judge whether the 
corrections to the preliminary coordinates will be significant, compared with the 
required or obtainable accuracy. 
6. Adjustment of rhomboids which depart further from the ideal shape. 
In order to study the limits up to which the solution presented in section 5 is