606
DOC.
581
JULY
1918
581. From
Felix Klein
Göttingen,
5
July 1918
Esteemed
Colleague,
I
still must write
you
about
my Monday
lecture.[1] I
have
the
proof you
had
intended:[2]
that the
Jo's form
a
contragradient
vector,
carried out
under the
restriction that
the world
tube
of
the closed
system
can
be
contained within
a
cylinder of finite
transverse
dimensions.[3]
The
accompanying figure
probably
does not need
any
explanation
(I
choose
the
co-
ordinate
system
in such
a way
that the
x4
axis
is
parallel
to the
cylinder’s
orientation;
a,
a'
are
any
two
points
linked
together
by
another
parallel;
they
define
the
vector
0, 0, 0,
Ax4).-
One
obtains
for
the
corresponding
integrals[4]
x4 =
C'
x4
=
C'
A"
=
J
ƒ
ƒ
ƒ
Uvodx1dx2dx3dx4
the
precise
scheme:[5]
0 0 0 Ax4
.
J1
0 0 0
Ax4
.
J2
0 0 0 Ax4
.
J3
0
0
0 Ax4
.
J4
Now I
introduce,
through
some
Lorentz
transformation,
instead
of
x,
new
x
coordinates
and
define
the
new
Avo's
by
means
of
a
cylinder
section whose
edges
x4
=
C,
and
=
C',
resp., again
go
through
the
points
a,
a1.
The
integra-
tion
domain
is
thereby visibly
different
from
the
original
one,
but
this
difference
bears
less
weight against
the
overall
ex-
tension,
the
larger Ax4
is
relative to
the
cylinder’s
cross
section.
We
conclude that
for
lim
Ax4
=
oo,
the
Avo’s
must
be
related
exactly
to the
Ava’s
as
the
Uvo’s are
to
the
Uvo's.[6]
Simultaneously,
Avo
=
Ja.Ax4.[7]
From
this then
follows
explicitly
the
desired correlation between Ja and
Jo,[8]
first