DOC.
71
PRINCETON LECTURES
271
PRE-RELATIVITY PHYSICS
If
we
multiply
the
equations
by
bßv
(compare (3a)
and
(5))
and
sum
for all
the
v’s,
we get
x'p
-
A'ß
=
\B'ß
where
we
have written
B'ß
=
5)
bß,BA'ß
=
bß,A,.
These
are
the
equations
of
straight
lines
with
respect
to
a
second
Cartesian
system
of
co-ordinates K'.
They
have the
same
form
as
the
equations
with
respect to
the
original
system
of
co-ordinates. It
is
therefore evident
that
straight
lines
have
a
significance
which
is
independent
of the
system
of
co-ordinates.
Formally,
this
depends
upon
the fact that the
quantities
(xv
-
Av)
-
\Bv
are
transformed
as
the
components
of
an
interval,
Axv.
The
ensemble of
three
quantities,
defined
for
every system
of
Cartesian
co-ordinates,
and which transform
as
the
com-
ponents
of
an
interval,
is
called
a
vector.
If
the
three
components
of
a
vector
vanish
for
one system
of
Cartesian
co-ordinates, they
vanish
for all
systems,
because the
equa-
tions
of transformation
are homogeneous.
We
can
thus
get
the
meaning
of the
concept
of
a
vector
without
referring
to
a
geometrical representation.
This behaviour
of
the
equations
of
a
straight
line
can
be
expressed by saying
that the
equation
of
a
straight
line
is
co-variant
with
respect
to
linear
orthogonal
transformations.
We shall
now
show
briefly
that there
are
geometrical
entities
which lead
to
the
concept
of
tensors.
Let
P0
be
the
centre
of
a
surface of the second
degree,
P
any point
on
the
surface,
and
£v
the
projections
of
the interval
P0P
upon
the
co-ordinate
axes.
Then the
equation
of
the
[11]