DETERMINATION OF ELLIPTIC ORBITS.
141
Δ
q
3
{\displaystyle \Delta q_{3}}
–.80780
–.04055
+.0025316
+ .0000031
q
3
{\displaystyle q_{3}}
+
{\displaystyle +}
3.24945
2.44165
2.40110
2.4036316
2.4036347
log
r
3
{\displaystyle \log r_{3}}
+
{\displaystyle +}
0.52600
.412217
.4057319
.4061394
.4061399
log
R
3
{\displaystyle \log R_{3}}
+
{\displaystyle +}
7.98439
8.325742
8.3451948
8.3439733
log
(
1
+
R
3
)
{\displaystyle \log(1+R_{3})}
+
{\displaystyle +}
.00417
.009099
.0095108
.0094843
log
P
‴
{\displaystyle \log P'''}
+
{\displaystyle +}
7.91715
8.357016
8.3817516
8.3801993
α
3
{\displaystyle \alpha _{3}}
−
{\displaystyle -}
1.17253
.987590
.9785152
.9790776
β
3
{\displaystyle \beta _{3}}
+
{\displaystyle +}
1.26749
.910305
.8924956
.8936069
γ
3
{\displaystyle \gamma _{3}}
−
{\displaystyle -}
.26373
.210171
.2075292
.2076940
α
‴
{\displaystyle \alpha '''}
−
{\displaystyle -}
.22847
.2222335
β
‴
{\displaystyle \beta '''}
+
{\displaystyle +}
.44441
.4390163
γ
‴
{\displaystyle \gamma '''}
+
{\displaystyle +}
.06690
.0650888
The values of
α
′
,
β
′
,
{\displaystyle \alpha ',\beta ',}
etc., furnish the basis for the computation of the following quantities:
a
1
=
−
.01254
{\displaystyle a_{1}=-.01254}
a
2
=
−
.03517
{\displaystyle a_{2}=-.03517}
a
3
=
−
.07232
{\displaystyle a_{3}=-.07232}
b
1
=
+
.01726
{\displaystyle b_{1}=+.01726}
b
2
=
−
.00525
{\displaystyle b_{2}=-.00525}
b
3
=
−
.00845
{\displaystyle b_{3}=-.00845}
c
1
=
−
.15746
{\displaystyle c_{1}=-.15746}
c
2
=
−
.08526
{\displaystyle c_{2}=-.08526}
c
3
=
−
.04050
{\displaystyle c_{3}=-.04050}
For we get three values sensibly identical. Adopting the mean, we set
G
=
.01006
.
{\displaystyle G=.01006.}
We also get
H
=
−
.00998
,
{\displaystyle H=-.00998,}
L
=
.02322
{\displaystyle L=.02322}
[ 1]
Taking the values of
α
1
,
α
2
,
{\displaystyle \alpha _{1},\alpha _{2},}
etc., from the columns under
III
1
,
III
2
,
III
3
,
{\displaystyle {\text{III}}_{1},{\text{III}}_{2},{\text{III}}_{3},}
we form the residuals
α
=
−
.06058
,
{\displaystyle \alpha =-.06058,}
β
=
−
.16692
,
{\displaystyle \beta =-.16692,}
γ
=
−
.05557
.
{\displaystyle \gamma =-.05557.}
From these, with the numbers last computed, we get
C
1
=
−
.65888
,
{\displaystyle C_{1}=-.65888,}
C
2
=
−
.76983
,
{\displaystyle C_{2}=-.76983,}
C
3
=
−
.79939
,
{\displaystyle C_{3}=-.79939,}
which might be used as corrections for our values of
q
1
,
q
2
,
q
3
.
{\displaystyle q_{1},q_{2},q_{3}.}
To get more accurate values for these corrections we set
Δ
q
2
=
C
2
−
6
10
L
(
Δ
q
2
)
2
,
{\displaystyle \Delta q_{2}=C_{2}-{\tfrac {6}{10}}L(\Delta q_{2})^{2},}
or
Δ
q
2
=
−
.76983
−
.01393
(
Δ
q
2
)
2
,
{\displaystyle \Delta q_{2}=-.76983-.01393(\Delta q_{2})^{2},}
which gives
Δ
q
2
=
−
.77826
.
{\displaystyle \Delta q_{2}=-.77826.}
The quadratic term diminishes the value of
Δ
q
2
{\displaystyle \Delta q_{2}}
by –.00843. Subtracting the same quantity from
C
1
{\displaystyle C_{1}}
and
C
2
{\displaystyle C_{2}}
we get
Δ
q
1
=
−
.66731
,
{\displaystyle \Delta q_{1}=-.66731,}
Δ
q
3
=
−
.80780
.
{\displaystyle \Delta q_{3}=-.80780.}
↑ It would have been better to omit altogether the oaloulation of
H
{\displaystyle H}
and
L
,
{\displaystyle L,}
if the small value of the latter coidd have been foreseen. In fact, it will be found that the terms containing
L
{\displaystyle L}
hardly improve the convergence, being smaller than quantities which have been neglected. Nevertheless, the use of these terms in this example will illustrate a process which in other cases may be beneficial.