356 DOC.71 PRINCETON LECTURES
THE
GENERAL
THEORY
M denotes the
sun’s
mass, centrally symmetrically placed
about
the
origin
of
co-ordinates;
the solution
(109a)
is
valid
only
outside
of this
mass,
where
all the
Tmv
vanish.
If the
motion
of
the
planet
takes
place
in
the
x1-x2
plane
then
[122]
we
must
replace (109a)
by
(109c) ds2
=
(l
-
^jdl2--d^-j{
-
r2d¡)2.
The calculation of the
planetary
motion
depends upon
equation
(90).
From the
first
of
equations
(110b)
and
[123]
(90)
we
get,
for
the
indices
1,
2, 3,
or,
if
we
integrate,
and
express
the result
in
polar
co-
ordinates,
(111)
r2~cls
=
constant.
From
(90),
for
n=4,
we
get
_
dH _d^l
\
dj2
d¿
^
ds2
f2
dxa
ds ds
ds2 f2
ds ds
From
this,
after
multiplication
by
f2
and
integration,
we
have
(112) f2dl/ds=constant.
In
(109c), (111)
and
(112)
we
have three
equations
between
the
four variables
s, r,
l and
f,
from
which the
motion
of
the
planet may
be
calculated
in
the
same
way
as
in classical
mechanics. The
most important
result
we
[96]
d
I
dxp
dXa\
~;
T -
~