DOC.
71
PRINCETON LECTURES 361
THE GENERAL THEORY
First,
we
consider
y44 as
small of the first
order. The
square
of
the
velocity
of
masses
moving
under
the influence
of the
gravitational
force
is
of the
same
order,
according
to
the
energy equation.
It
is
therefore
logical
to regard
the
velocities of the
material
particles
we are
considering,
as
well
as
the
velocities of
the
masses
which
generate
the
field,
as
small,
of the
order
1/2.
We shall
now
carry out
the
approximation
in the
equations
that
arise from
the
field
equations
(101)
and
the
equations
of
motion
(90)
so
far
as
to
consider
terms,
in the second
member
of
(90),
that
are
linear
in those velocities.
Further,
we
shall
not put
ds
and
dl
equal
to
each
other, but,
corresponding
to
the
higher approximation,
we
shall
put
ds
=
V£44
dl
=
^1
-
dl.
From
(90)
we
obtain,
at
first,
(116)
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From
(101)
we
get,
to
the
approximation sought for,
(117)
K
frdVo
=
~Y22
=
~Y33
=
~Y44
=
j
J
r
~Y4a
=
-
dV0
`YaP
=
0
in
which,
in
(117),
a
and
ß
denote the
space
indices
only.
On the
right-hand
side of
(116)
we
can
replace
1
+
y44/2
by
1
and
-
TaBu
by
[aBu].
It
is easy
to
see,
in
addition,
that
to
this
degree
of
approximation
we
must
put
[101]
[131]