5 4 D O C . 9 E N E R G Y C O N S E RVAT I O N
On the other hand, since can be made into by continuous changes, our gen-
eral theorem of invariance demands for the
.
(12)
(11) and (12 together imply the vanishing of
and .
The vanishing of and can be proven in analogy by the continuous change
of the coordinates of a system that is connected to by the substitution
. (10a)
Now we only have to prove that the substitutions (10) and (10a) can be generated
by a continuous change of the coordinate system. We can limit the consideration to
a three-dimensional sphere, leaving the -coordinate aside.
In a four-dimensional space of the , the sphere shall satisfy the equation
.
In a four-dimensional Euclidean space we connect with these Cartesian coordi-
nates the spherical coordinates through the formula
.
(13)
K K
′
Jσ
J1 ′
J1}
=
J2 ′ J2 =
J1 J2
J1 J3
K
′
K
ϑ′
1
π ϑ1 – =
ϑ′2 ϑ2 =
ϑ′3 2π –ϑ3 =
t′ t =
t
uν
u2
1
u2
2 u2
3
u2
4
+ + +
R2
=
u1 R ϑ1 cos =
[p. 455]
u2 R ϑ1 ϑ2 cos sin =
u3 R ϑ1 ϑ2 sin ϑ3 cos sin =
u4 R ϑ1 ϑ2 sin ϑ3 sin sin =