*9
or ev GfrG«v U^KP
whence, as Gx$ gap gPY 8Y I if a f, and o if a y
e«=U°K. (9)
In other words:
The corrections e of the original problem can be regarded as
corrections to uncorrelated observations of equal weights, to be
submitted to the condition-equations
UpRa=Up.
These equations are now solved by the orthogonalization method.
VI. Example
The sequence of the calculations will be shown. A triangulation
is given as in fig. 6. All directions are observed with equal weights
and are assumed to be uncorrelated. In this case the cofactors-
tensor is:
g«P=i (a p) and g°$= o (a (J) a (3 I 62.
The adjustment will be performed in three steps. In all there are
17 triangle-conditions and 4 side-conditions.
a) The first step consists of adjusting 11 angle-conditions.
A chain of triangles ischosen in the net e.g.no 1,..,11 incl. Its direc
tions are numbered according to the system as under III, A. From
formula II, 3 is derived:
e1=+K1
e2 Kx
e3 -\-Kx K2 etc.
The correlatives are immediately known as the inverse matrix of the
normal-equations is given by the rules of III, B and so is the co-
factors-tensor. (See III, C, D, and E.) The corrections are computed
and applied.
b) In the second phase the once corrected directions have now
unequal weights and are correlated. This is expressed by the above
mentioned tensor.
The two triangle-equations 12 and 13 and three side-equations
are adjusted. The conditions are given by
up P<* up.
This set of equations is submitted to a linear transformation (see V)
whence
XJo ra p.
To find the corrections e these equations are solved by the orthog-
a p
a o
a o
„0.3 mp 7^0. U9 =up R<*
o a o a
a