DOC.
185
JANUARY
1916 183
all
on
the
papers
but
shall calculate
everything
out for
you.[2]
Then,
if
anything
should remain
incomprehensible,
the
gap
can
be
easily
filled.-
1)
Lagrangian
form
of
the
equations.
Statement:
Let
V-g
=
1.
In
addition,
let L
=
gor{aBa}{TaB}.
Then
follows,
if
£
is
conceived
of
as a
function
of
the
gaT's
and
goT
=
dgoT/dxo:[3]
8L
dg°
aß
a
ra
ß
dL
dgoT
aß
a
(1)
Comment: I
always
omit
the
summation
symbol.
An index must
always
be
summed when it
appears
twice.[4]
Proof: From
differentiating
£-always considered
as a
funct.
of
gor
&
goT-follows
aß
I
I
ra
dL
=
a
ß
dgaT
+
2g°
i\A
t
(Two
terms
differing only
in index
name are summarized).
From
this
follows
furthermore
from
goTd
ra Ta
Ta
ß
=
d
\g
ß ß
dg0
dL
=
-dg"
(
f
Ta
+ 2
aß
Ta
ß
a
ß
=
-dgCT{
} {
}
+
2
aß
d
(
g°TgßX
Ta
a A
In
addition
aß
a
ßa
a
(a)
Taking
into
consideration
that the
second term does
not
change
when the
sum
of
the
indices
a
& ß
are
exchanged simultaneously
with
A
and
r,
the
second term
is
then
also
equal
to
aß
a
d
gaTg‘
or
ß\
Ta
A
+
A
a
T