116 DOC.
24
PERIHELION MOTION OF MERCURY
where
al
and
a2
signify
the
roots
of the
equation
2
A a
. ,
rv
+
r
x

x2 + ax3
=
0
B2 B2
and
closely
correspond
to
the
neighboring
roots
of the
equation
that arises from this
one
by
the omission of the last
term.
Thus, it
can
be
established with the
precision
demanded of
us
that
[17]
«2 dx
0
=
[l
+
a(a,+a2)]f

J"
a2)(l

ax)
[16]
or
upon
expansion
of
(1

ax)1/2,
f)
=
[1
+ a(ai +
a2)]
*is/(xal)(xa2)
[18]
The
integration yields
1
+*(«!
+oc2)
or
if
we
consider that
a1
and
a2 signify
the
reciprocal
values
of the maximum and minimum
distances,
respectively,
from
the
sun,
p
=
nl
1
+

a
(12)
2
a(
1 
e2)
Therefore,
after
a complete
orbit,
the
perihelion
advances
by
e
=
37:
a
(13)
a(l

e2)
in
the
sense
of the orbital
motion,
where the
semimajor
axis
is
denoted
by a
and the
eccentricity
by e.
If
we
introduce the
[3]
orbital
period
T
(in
seconds),
we
obtain
e
=
24rc3
(14)
T2c2(l

e2)'
where
c
denotes the
velocity
of
light
in units of
cm
sec1.
The
calculation
yields,
for the
planet Mercury,
a
perihelion
advance of 43"
per century,
while the astronomers
assign
45" ±
5"
per century as
the
unexplained difference
between
observations and the Newtonian
theory.
This
theory
therefore
agrees
completely
with the observations.
For Earth and
Mars,
the
astronomers assign,
respectively,
forward motions of
11"
and 9"
per century,
while
our
formula
yields,
respectively,
4" and
1"
per century.
Nevertheless,
a
small value
seems
to
be
proper
to these
assignments
because
of
the small
eccentricities of
the
orbits of
these
planets. A
more
certain confirmation of the
perihelion
motion
will
be
made
by
determining
the
product
of
the
motion with the
eccentricity.
If
we
consider these
quantities assigned by Newcomb,
edn/dt
Mercury 8.48"
±
0.43
Venus
0.05
±
0.25
Earth
0.10
±
0.13
Mars
0.75
±
0.35,
for which
I
thank Dr.
Freundlich,
then
we
obtain the
im
pression
that the advance
of
the
perihelion is,
after
all,
demonstrated
really only
for
Mercury. However, I
prefer
to
relinquish
a
final decision
to
the astronomical
specialists.
1.
In
a forthcoming
communication
it will be
shown that
this
hypothesis
is
unnecessary.
It
is
because such
a
choice of
reference frame
is possible
that the determinant
guv
takes
on
the value
1.
The
following investigation is independent
of
this
choice.
2. E.
Freundlich
recently
wrote
a noteworthy
article
on
the
impossibility
of
satisfactorily explaining
the
anomalies
in the motion
of
Mercury on
the basis
of
the
Newtonian
theory (Astronomische
Nachrichten
201,
49
[1915]).
[5]