Components of
S
‴
{\displaystyle {\mathfrak {S}}'''}
P
‴
=
3
R
3
q
3
(
1
+
R
3
)
r
3
2
{\displaystyle P'''={\frac {3R_{3}q_{3}}{(1+R_{3})r_{3}^{2}}}}
α
‴
=
A
3
ξ
3
+
A
3
ξ
3
R
3
−
P
‴
α
3
{\displaystyle \alpha '''=A_{3}\xi _{3}+A_{3}\xi _{3}R_{3}-P'''\alpha _{3}}
}
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}
III
‴
{\displaystyle {\text{III}}'''}
β
‴
=
A
3
η
3
+
A
3
η
R
3
−
P
‴
β
3
{\displaystyle \beta '''=A_{3}\eta _{3}+A_{3}\eta _{}R_{3}-P'''\beta _{3}}
γ
‴
=
A
3
ζ
3
+
A
3
ζ
3
R
3
−
P
‴
γ
3
{\displaystyle \gamma '''=A_{3}\zeta _{3}+A_{3}\zeta _{3}R_{3}-P'''\gamma _{3}}
The computer is now to assume any reasonable values either of the geocentric distances,
ρ
1
,
ρ
2
,
ρ
3
,
{\displaystyle \rho _{1},\rho _{2},\rho _{3},}
r
1
,
r
2
,
r
3
{\displaystyle r_{1},r_{2},r_{3}}
By equations
III
1
,
III
′
.
{\displaystyle {\text{III}}_{1},{\text{III}}'.}
By equations
III
2
,
III
″
.
{\displaystyle {\text{III}}_{2},{\text{III}}''.}
By equations
III
3
,
III
‴
.
{\displaystyle {\text{III}}_{3},{\text{III}}'''.}
q
1
{\displaystyle q_{1}}
q
2
{\displaystyle q_{2}}
q
3
{\displaystyle q_{3}}
log
r
1
{\displaystyle \log r_{1}}
log
r
2
{\displaystyle \log r_{2}}
log
r
3
{\displaystyle \log r_{3}}
log
R
1
{\displaystyle \log R_{1}}
log
R
2
{\displaystyle \log R_{2}}
log
R
3
{\displaystyle \log R_{3}}
log
(
1
+
R
1
)
{\displaystyle \log(1+R_{1})}
log
(
1
+
R
2
)
{\displaystyle \log(1+R_{2})}
log
(
1
+
R
3
)
{\displaystyle \log(1+R_{3})}
log
P
′
{\displaystyle \log P'}
log
P
″
{\displaystyle \log P''}
log
P
‴
{\displaystyle \log P'''}
α
1
{\displaystyle \alpha _{1}}
α
2
{\displaystyle \alpha _{2}}
α
3
{\displaystyle \alpha _{3}}
β
1
{\displaystyle \beta _{1}}
β
2
{\displaystyle \beta _{2}}
β
3
{\displaystyle \beta _{3}}
γ
1
{\displaystyle \gamma _{1}}
γ
2
{\displaystyle \gamma _{2}}
γ
3
{\displaystyle \gamma _{3}}
α
′
{\displaystyle \alpha '}
α
″
{\displaystyle \alpha ''}
α
‴
{\displaystyle \alpha '''}
β
′
{\displaystyle \beta '}
β
″
{\displaystyle \beta ''}
β
‴
{\displaystyle \beta '''}
γ
′
{\displaystyle \gamma '}
γ
″
{\displaystyle \gamma ''}
γ
‴
{\displaystyle \gamma '''}
IV.
Calculations relating to differential coefficients.
Components of
S
″
×
S
‴
{\displaystyle {\mathfrak {S}}''\times {\mathfrak {S}}'''}
Components of
S
‴
×
S
′
{\displaystyle {\mathfrak {S}}'''\times {\mathfrak {S}}'}
Components of
S
′
×
S
″
{\displaystyle {\mathfrak {S}}'\times {\mathfrak {S}}''}
a
1
=
β
″
γ
‴
−
γ
″
β
‴
{\displaystyle a_{1}=\beta ''\gamma '''-\gamma ''\beta '''}
a
1
=
β
‴
γ
′
−
γ
‴
β
′
{\displaystyle a_{1}=\beta '''\gamma '-\gamma '''\beta '}
a
3
=
β
′
γ
″
−
γ
′
β
″
{\displaystyle a_{3}=\beta '\gamma ''-\gamma '\beta ''}
b
1
=
γ
″
α
‴
−
α
″
γ
‴
{\displaystyle b_{1}=\gamma ''\alpha '''-\alpha ''\gamma '''}
b
2
=
γ
‴
α
′
−
α
‴
γ
′
{\displaystyle b_{2}=\gamma '''\alpha '-\alpha '''\gamma '}
b
3
=
γ
′
α
″
−
α
′
γ
″
{\displaystyle b_{3}=\gamma '\alpha ''-\alpha '\gamma ''}
c
1
=
α
″
β
‴
−
β
″
α
‴
{\displaystyle c_{1}=\alpha ''\beta '''-\beta ''\alpha '''}
c
2
=
α
‴
β
′
−
β
‴
α
′
{\displaystyle c_{2}=\alpha '''\beta '-\beta '''\alpha '}
c
3
=
α
′
β
″
−
β
′
α
″
{\displaystyle c_{3}=\alpha '\beta ''-\beta '\alpha ''}
These computations are controlled by the agreement of the three values of
G
.
{\displaystyle G.}
The following are not necessary except when the corrections to be made are large:
H
=
(
F
2
S
‴
S
′
)
=
a
2
ξ
2
+
b
2
η
2
+
c
2
ζ
2
L
=
1
q
2
(
1
+
H
G
)
(
1
−
5
q
2
2
r
2
2
)
−
R
2
H
q
2
G
(
1
+
q
2
2
r
2
2
)
{\displaystyle {\begin{aligned}H&=({\mathfrak {F}}_{2}{\mathfrak {S}}'''{\mathfrak {S}}')=a_{2}\xi _{2}+b_{2}\eta _{2}+c_{2}\zeta _{2}\\L&={\frac {1}{q_{2}}}\left(1+{\frac {H}{G}}\right)\left(1-5{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)-{\frac {R_{2}H}{q_{2}G}}\left(1+{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)\end{aligned}}}
