442
The Theory of Aether and Electrons in the
that is to say, at the place
(
ξ
1
+
Δ
ξ
cosh
a
,
η
1
+
Δ
η
,
ζ
1
+
Δ
ζ
)
{\displaystyle (\xi _{1}+\Delta \xi \cosh a,\qquad \eta _{1}+\Delta \eta ,\qquad \zeta _{1}+\Delta \zeta )}
at the instant {{Wikimath|(t 1 - sinh a . Δξ /c ). Thus at the instant t 1 , this charge will occupy the position
(
ξ
1
+
Δ
ξ
cosh
a
+
sinh
a
.
Δ
ξ
.
v
x
1
/
c
,
η
1
+
Δ
η
+
sinh
a
.
Δ
ξ
.
v
y
1
/
c
{\displaystyle (\xi _{1}+\Delta \xi \cosh a+\sinh a.\Delta \xi .v_{x_{1}}/c,\qquad \eta _{1}+\Delta \eta +\sinh a.\Delta \xi .v_{y_{1}}/c\qquad }
ζ
1
+
Δ
ζ
+
sinh
a
.
Δ
ξ
.
v
a
1
/
c
)
{\displaystyle \zeta _{1}+\Delta \zeta +\sinh a.\Delta \xi .v_{a_{1}}/c)}
The charges corresponding to those in the original system which were at the instant t contained in a volume Δξ Δη Δζ will therefore in the derived system at the instant t 1 , occupy a volume
|
cosh
a
+
sinh
a
.
v
x
1
/
c
.
0
0
sinh
a
.
v
y
1
/
c
1
0
sinh
a
.
v
z
1
/
c
0
1
|
.
Δ
ξ
Δ
η
Δ
ζ
{\displaystyle \left|{\begin{matrix}\cosh a&+&\sinh a.v_{x_{1}}/c.&0&0\\\ &\ &\sinh a.v_{y_{1}}/c&1&0\\\ &\ &\sinh a.v_{z_{1}}/c&0&1\\\end{matrix}}\right|.\Delta \xi \Delta \eta \Delta \zeta }
or,
(
cosh
a
+
sinh
a
.
v
x
1
/
c
)
Δ
ξ
Δ
η
Δ
ζ
{\displaystyle (\cosh a+\sinh a.v_{x_{1}}/c)\Delta \xi \Delta \eta \Delta \zeta }
.
Thus if ρ 1 denote the volume-density of electric charge in the transformed system, we shall have
ρ
1
(
cosh
a
+
sinh
a
.
v
x
1
/
c
)
=
ρ
{\displaystyle \rho _{1}(\cosh a+\sinh a.v_{x_{1}}/c)=\rho }
;
this equation expresses the connexion between ρ 1 and ρ . We have moreover
v
x
=
∂
x
∂
x
1
v
x
1
+
∂
x
∂
y
1
v
y
1
+
∂
x
∂
z
1
v
z
1
+
∂
x
∂
t
1
∂
t
∂
x
1
v
x
1
+
∂
t
∂
y
1
v
y
1
+
∂
t
∂
z
1
v
z
1
+
∂
t
∂
t
1
=
e
tanh
a
+
v
x
1
sech
a
cosh
a
+
v
x
1
e
−
1
sinh
a
{\displaystyle {\begin{matrix}v_{x}&=&{\frac {{\frac {\partial x}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial x}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial x}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial x}{\partial t_{1}}}}{{\frac {\partial t}{\partial x_{1}}}v_{x_{1}}+{\frac {\partial t}{\partial y_{1}}}v_{y_{1}}+{\frac {\partial t}{\partial z_{1}}}v_{z_{1}}+{\frac {\partial t}{\partial t_{1}}}}}\\\ &=&e\tanh a+{\frac {v_{x_{1}}{\text{sech }}a}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}\end{matrix}}}
and similarly
v
y
=
v
y
1
cosh
a
+
v
x
1
e
−
1
sinh
a
{\displaystyle v_{y}={\frac {v_{y_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}}
,
and
v
x
=
v
x
1
cosh
a
+
v
x
1
e
−
1
sinh
a
{\displaystyle v_{x}={\frac {v_{x_{1}}}{\cosh a+v_{x_{1}}e^{-1}\sinh a}}}
.