VIII.
For the second hypothesis.
δ
τ
1
{\displaystyle \delta \tau _{1}}
=
.0057613
k
(
ρ
2
−
ρ
3
)
{\displaystyle =.0057613k(\rho _{2}-\rho _{3})}
(aberration-constant after Struve.)
δ
τ
2
{\displaystyle \delta \tau _{2}}
=
.0057613
k
(
ρ
1
−
ρ
2
)
{\displaystyle =.0057613k(\rho _{1}-\rho _{2})}
log
(
.0057613
k
)
=
5.99610
{\displaystyle \log(.0057613k)=5.99610}
Δ
log
τ
1
{\displaystyle \Delta \log \tau _{1}}
=
log
τ
1
−
log
(
τ
1
calc.
−
δ
τ
1
)
{\displaystyle =\log \tau _{1}-\log(\tau _{1{\text{ calc.}}}-\delta \tau _{1})}
Δ
log
τ
3
{\displaystyle \Delta \log \tau _{3}}
=
log
τ
3
−
log
(
τ
3
calc.
−
δ
τ
3
)
{\displaystyle =\log \tau _{3}-\log(\tau _{3{\text{ calc.}}}-\delta \tau _{3})}
Δ
log
(
τ
1
τ
3
)
{\displaystyle \Delta \log(\tau _{1}\tau _{3})}
=
Δ
log
τ
1
+
Δ
log
τ
3
{\displaystyle =\Delta \log \tau _{1}+\Delta \log \tau _{3}}
Δ
log
τ
1
τ
3
{\displaystyle \Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
=
Δ
log
τ
1
−
Δ
log
τ
3
{\displaystyle =\Delta \log \tau _{1}-\Delta \log \tau _{3}}
Δ
log
A
1
{\displaystyle \Delta \log A_{1}}
=
−
A
3
Δ
log
τ
1
τ
3
{\displaystyle =-A_{3}\Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
Δ
log
A
3
{\displaystyle \Delta \log A_{3}}
=
−
A
1
Δ
log
τ
1
τ
3
{\displaystyle =-A_{1}\Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
Δ
log
B
1
{\displaystyle \Delta \log B_{1}}
=
Δ
log
(
τ
1
τ
3
)
−
τ
1
2
+
τ
3
2
12
B
1
Δ
log
τ
1
τ
3
{\displaystyle =\Delta \log(\tau _{1}\tau _{3})-{\frac {\tau _{1}^{2}+\tau _{3}^{2}}{12B_{1}}}\Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
Δ
log
B
2
{\displaystyle \Delta \log B_{2}}
=
Δ
log
(
τ
1
τ
3
)
+
τ
1
2
−
τ
3
2
12
B
2
Δ
log
τ
1
τ
3
{\displaystyle =\Delta \log(\tau _{1}\tau _{3})+{\frac {\tau _{1}^{2}-\tau _{3}^{2}}{12B_{2}}}\Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
Δ
log
B
3
{\displaystyle \Delta \log B_{3}}
=
Δ
log
(
τ
1
τ
3
)
+
τ
1
2
+
τ
3
2
12
B
3
Δ
log
τ
1
τ
3
{\displaystyle =\Delta \log(\tau _{1}\tau _{3})+{\frac {\tau _{1}^{2}+\tau _{3}^{2}}{12B_{3}}}\Delta \log {\frac {\tau _{1}}{\tau _{3}}}}
These corrections are to be added to the logarithms of
A
1
,
A
3
,
B
1
,
B
2
,
B
3
{\displaystyle A_{1},A_{3},B_{1},B_{2},B_{3}}
III
1
,
III
2
,
III
3
{\displaystyle {\text{III}}_{1},{\text{III}}_{2},{\text{III}}_{3}}
q
1
,
q
2
,
q
3
{\displaystyle q_{1},q_{2},q_{3}}
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
A
1
,
A
3
{\displaystyle A_{1},A_{3}}
A
1
+
A
3
=
1
,
{\displaystyle A_{1}+A_{3}=1,}
Δ
log
A
1
,
Δ
log
A
3
{\displaystyle \Delta \log A_{1},\Delta \log A_{3}}
A second correction of equations
III
1
,
III
2
,
III
3
{\displaystyle {\text{III}}_{1},{\text{III}}_{2},{\text{III}}_{3}}
IX.
Determination of the ellipse.
It is supposed that the values of
α
1
,
β
1
,
γ
1
,
{\displaystyle \alpha _{1},\beta _{1},\gamma _{1},}
α
2
,
β
2
,
γ
2
,
{\displaystyle \alpha _{2},\beta _{2},\gamma _{2},}
α
3
,
β
3
,
γ
3
,
{\displaystyle \alpha _{3},\beta _{3},\gamma _{3},}
r
1
,
r
2
,
r
3
,
{\displaystyle r_{1},r_{2},r_{3},}
R
1
,
R
2
,
R
3
,
{\displaystyle R_{1},R_{2},R_{3},}
s
1
,
s
2
,
s
3
,
{\displaystyle s_{1},s_{2},s_{3},}
have been computed by equations
III
1
,
III
2
,
III
3
{\displaystyle {\text{III}}_{1},{\text{III}}_{2},{\text{III}}_{3}}
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
s
1
,
s
2
,
s
3
{\displaystyle s_{1},s_{2},s_{3}}
