SE
RE RSS
= ea
80 ADJUSTMENT OF RHOMBOIDS, ACKERMANN
4. The logarithmic discrepancy.
The original form (3.1) of the condition equation can be written directly as eoe =
— 1, where the indices C and D (indieating the way of computation) distinguish the two
different values of the distance e.
On introducing the
“linear discrepancy” Ade — e,
en (4.1)
the condition equation becomes
(1 +27) TE (4.2)
\ ep,
Ae
The logarithm of this equation, being In E +“ 2 ) — 0, can be expanded into:
D
Ae ( le |
€ i rom 20 (4.3
D N
As the term /e/e, wil be a small value, hardly exceeding 10-3, only the first
term of the series is ever needed. Moreover the value ep may always be replaced by any
approximate value for e, as an accuracy of the order of a few percent is sufficient.
As the above derivation corresponds, step by step, to that of the preceeding section,
it is obvious, that the equation (4.3), which is sufficiently accurate with its first term
only, is the same as (3.2). This means that, on computing Ae with direct observations,
the resulting discrepancy
f= —— (4.4)
is identical with the logarithmic discrepancy t, as defined in (3.3) and can fully replace
it there.
It can be stated that the logarithmic discrepancy t (in terms of natural logarithms)
is equal to the discrepancy in scale, resulting from the two possible computations of the
distance e. As this computation can be carried out with the ordinary means, without using
logarithmie tables, the first of the aims mentioned initially is thus achieved.
For all further’ considerations in this paper, which refer to the equation (34), it
should be assumed, that the logarithmic discrepancy t is computed according to equation
(4.4) instead of equation (3.3).
5. Adjustment of approximately ideal-shaped rhomboids.
In the linearized condition equation (3.4) the coefficients of the corrections v depend
on the angles or in other words on the shape of the rhomboid. As the ideal shape of a
rhomboid is square an obvious simplification lies at hand, i.e. to assume for these coef-
fieients those values valid for the ideal case. This simplification promises to be successful
for approximately square-shaped rhomboids, because it is well established from expe-
rience that the coefficients of differentia! quantities (and the corrections v are mostly
smaller than 1*9) need no great accuracy.
y. With the simplified coefficients the condition
equation is
i On KARL Day Bye Oi 04 Lay iB Ln
2v, 1v, 2v. 2v4 -F 2v, LU, + 2v, — 2v, ta
das ï
9— 209, —t=0 (5.1)
Hence the normal equation will be, assuming equal weights, and no correlation, for
the ten observations
de
e
64 K t c
