108 DOCUMENT 67 MARCH 1915
That
is
indeed the
case.
Therefore it is
necessary
to
give a
concrete
example
in which
not all Fuv vanish
as a
result
of
some
admissible transformation from
a
Euclidean ds2
(contrary
to what covariance would
require).
This
is
how
we
go
about
it.
We start from
a
coordinate
system
in which
ds2
has the canonical
Euclidean
form
ds2
=
dx
\
+
dx\ +
dx2
+
dx\
-
duv),
and
we proceed
to
perform
an
infinitesimal
transformation,
putting
/
X
"
=
H
=
*n
+ 7
v
.
(1)
where the
yu
designate
infinitesimal
(a priori any)
functions
of
x.
Putting
duv
+
huv
(2)
for
the guv
relative to
the
new
variables
x',
we
have
(tyi
.
â’v
huv
l.dJCv
+
abcM
(3)
Moreover,
with
your
expression
(78)
for
H,[3]
and,
of
course,
always omitting
terms
of
higher order,
d(fíVg)
=
ldVr
dgfv
2
dxa
It follows
(changing g
into
-g)[4]
a
-¿(HJg)
y
d
fdHjg)_
1
dg(nv
ZjadX(j[dg^v)J
22^
’
(4)
where for
brevity we designate
with
A2
the differential
operator (Laplace’s)
y
-il
Having
stated this
general
theorem, we
note
that
the
infinitesimal
transformation
(1)
will be
adapted
if
the
y’s satisfy
the
four
equations
(65a)[5]
,2
B
=
y
_-
fp(va)^^'^¿))=
0
U
-^«av
dx0dx0
\8
dg^v
)
which
in
the
present
case
become
ly
7¿us
"
,
^-A7/ji1v
-
0.
2^vdr
2
t1''
(5)
Because
of
expressions (3)
for the huv, these
are
linear
equations, homogeneous
and
of
the
fourth
order
in
y.
They are
therefore satisfied
if
for the
y’s
we
take
some arbitrary polyno-
mials
of
the
third
degree
in
x.
We
assume
in
particular
3V
=
,