∂ h z ∂ y − ∂ h y ∂ z = 1 c ( ∂ ∂ t − w ∂ ∂ x ) d x + 1 c ϱ ( w + u x ) {\displaystyle {\frac {\partial {\mathfrak {h}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {h}}_{y}}{\partial z}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{x}+{\frac {1}{c}}\varrho \left(w+{\mathfrak {u}}_{x}\right)} ,
∂ h x ∂ z − ∂ h z ∂ x = 1 c ( ∂ ∂ t − w ∂ ∂ x ) d y + 1 c ϱ u y {\displaystyle {\frac {\partial {\mathfrak {h}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {h}}_{z}}{\partial x}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{y}+{\frac {1}{c}}\varrho {\mathfrak {u}}_{y}} ,
∂ h y ∂ x − ∂ h x ∂ y = 1 c ( ∂ ∂ t − w ∂ ∂ x ) d z + 1 c ϱ u z {\displaystyle {\frac {\partial {\mathfrak {h}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {h}}_{x}}{\partial y}}={\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {d}}_{z}+{\frac {1}{c}}\varrho {\mathfrak {u}}_{z}} ,
∂ d z ∂ y − ∂ d y ∂ z = − 1 c ( ∂ ∂ t − w ∂ ∂ x ) h x {\displaystyle {\frac {\partial {\mathfrak {d}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{y}}{\partial z}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{x}} ,
∂ d x ∂ z − ∂ d z ∂ x = − 1 c ( ∂ ∂ t − w ∂ ∂ x ) h y {\displaystyle {\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {d}}_{z}}{\partial x}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{y}} ,
∂ d y ∂ x − ∂ d x ∂ y = − 1 c ( ∂ ∂ t − w ∂ ∂ x ) h z {\displaystyle {\frac {\partial {\mathfrak {d}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}=-{\frac {1}{c}}\left({\frac {\partial }{\partial t}}-w{\frac {\partial }{\partial x}}\right){\mathfrak {h}}_{z}} ,
f x = d x + 1 c ( u y h z − u z h y ) {\displaystyle {\mathfrak {f}}_{x}={\mathfrak {d}}_{x}+{\frac {1}{c}}\left({\mathfrak {u}}_{y}{\mathfrak {h}}_{z}-{\mathfrak {u}}_{z}{\mathfrak {h}}_{y}\right)} ,
f y = d y − 1 c w h z + 1 c ( u z h x − u x h z ) {\displaystyle {\mathfrak {f}}_{y}={\mathfrak {d}}_{y}-{\frac {1}{c}}w{\mathfrak {h}}_{z}+{\frac {1}{c}}\left({\mathfrak {u}}_{z}{\mathfrak {h}}_{x}-{\mathfrak {u}}_{x}{\mathfrak {h}}_{z}\right)} ,
f z = d z + 1 c w h y + 1 c ( u x h y − u y h x ) {\displaystyle {\mathfrak {f}}_{z}={\mathfrak {d}}_{z}+{\frac {1}{c}}w{\mathfrak {h}}_{y}+{\frac {1}{c}}\left({\mathfrak {u}}_{x}{\mathfrak {h}}_{y}-{\mathfrak {u}}_{y}{\mathfrak {h}}_{x}\right)} .
§ 4. We shall further transform these formulae by a change of variables. Putting
and understanding by l another numerical quantity, to be determined further on, I take as new independent variables
and I define two new vectors d ′ {\displaystyle {\mathfrak {d}}'} and h ′ {\displaystyle {\mathfrak {h}}'} by the formulae
h x ′ = 1 l 2 h x , h y ′ = k l 2 ( h y + w c d z ) , h z ′ = k l 2 ( h z − w c d y ) {\displaystyle {\mathfrak {h}}_{x}^{'}={\frac {1}{l^{2}}}{\mathfrak {h}}_{x},\quad {\mathfrak {h}}_{y}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {h}}_{y}+{\frac {w}{c}}{\mathfrak {d}}_{z}\right),\quad {\mathfrak {h}}_{z}^{'}={\frac {k}{l^{2}}}\left({\mathfrak {h}}_{z}-{\frac {w}{c}}{\mathfrak {d}}_{y}\right)} ,
for which, on account of (3), we may also write