264
THE
RELATIVITY PRINCIPLE
[25]
u
~
X
ul
+
V
X
un =
y
u
z
1
+
nx
1
V2
"
72"
1
+
vux
cl
1
4
yl
I
y
1 +
X
I
£
(3)
The law
of
the
parallelogram
of velocities thus
holds
only
in
first
approximation.
If
we
set
u2
-
u1
+
u2
+
u2
x
y
z
U'2
=
tt'2
+
w'2
+
u12
x
y
z
and
denote
by a
the
angle
between
the x'-axis
(v)
and
the point's
direction
of motion
relative
to S'
(w'),
we
will
have
[26]
u
-
(v2-i-u'2÷2vu'cosa)-
vu' sin
~
~2
2
1
I
+ v?t
cos
If the
two
velocities
(v
and u') have
the
same
direction,
we
have
u v
+
u
!
1
4-
VU
1
+
"72
It follows
from
this
equation
that the addition of
two
velocities
smaller than
c
always
results in
a
velocity smaller than
c;
i.e., if
one
sets
v
=
c
-
k,
u'
=
c
-
A,
where
k
and
A
are
positive
and
smaller than
c,
then
2c
-
k
-
A
,
"
U =
C
t~t~
c
.
2c
-
k
- A
+
^