272 DOC. 71 PRINCETON LECTURES
PRE-RELATIVITY PHYSICS
surface
is
2
=
1.
In
this,
and
in
analogous cases,
we
shall
omit the
sign
of
summation,
and understand that
the summation
is
to
be
carried
out
for those indices
that
appear twice.
We thus
[18]
write
the
equation
of
the
surface
auvEuEv
=
1.
The
quantities
auv
determine the
surface
completely,
for
a
given position
of the centre,
with
respect
to
the
chosen
system
of
Cartesian
co-ordinates. From
the known law
of transformation
for the
£v
(3a)
for linear
orthogonal
transformations,
we
easily
find the law
of
transformation
for
the
auv*:
=
boubTvauv.
This
transformation
is
homogeneous
and
of
the
first
degree
in
the
auv.
On
account
of
this
transformation,
the
auv
are
called
components
of
a
tensor
of
the
second
rank
(the
latter
on
account
of the
double
index).
If
all
the
com-
ponents,
auv,
of
a
tensor
with
respect to
any system
of
Cartesian co-ordinates
vanish,
they
vanish with
respect to
every
other Cartesian
system.
The
form
and
the
position
of the
surface of
the
second
degree
is
described
by
this
tensor (a).
Tensors of
higher
rank
(number
of
indices)
may
be
defined
analytically.
It
is
possible
and
advantageous
to
regard
vectors
as
tensors
of
rank
1,
and invariants
(scalars)
as
tensors
of rank
0.
In
this
respect,
the
problem
of the
theory
of invariants
may
be
so
formulated:
according to
*The
equation
a'ort'rt'r
=
1
may, by (5),
be
replaced by a'orbuobvttUÍt
=
1,
from which the result stated
immediately
follows.
[12]