Now upon the basis of the equations (55) and (56), and referring back to
the expression (82) for L, and from 57) we obtain the following
expressions as components of N,—
(92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_}
+ (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_}
+ Ω₄ ∂ω₄/∂_x__{_h_})
for _h_ = 1, 2, 3, 4.
Now if we make use of (59), and denote the space-vector which has Ω₁,
Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third
component of 92) can be expressed in the form
(93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),
The round bracket denoting the scalar product of the vectors within it.
§ 14. The Ponderomotive Force.[28]
Let us now write out the relation K = lor S = -_s_F + N in a more
practical form; we have the four equations
(94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ -
∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}
- ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_x_),
(95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ -
∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}
- ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_y_),
(96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ -
∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄
- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_z_),
(97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ -
∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} +
_s__{_z_}E_{_z_}
- ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_t_).
It is my opinion that when we calculate the ponderomotive force which
acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it
has got, _x_, _y_, _z_ components as the first three components of the
space-time vector
K + (ωK)ω,
This vector is perpendicular to ω; the law of Energy finds its
expression in the fourth relation.
The establishment of this opinion is reserved for a separate tract.
In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK =
0, and we obtain the ordinary equations in the theory of electrons.
Footnote 9:
_Vide_ Note 1.
Footnote 10:
Note 2.
Footnote 11:
_Vide_ Note 3.
Footnote 12:
_Vide_ Note 4.
Footnote 13:
Note 5.
Footnote 14:
See notes on § 8 and 10.
Footnote 15:
See note 9.
Footnote 16:
See Note.
Footnote 17:
Vide Note.
Footnote 18:
Just as beings which are confined within a narrow region surrounding a
point on a spherical surface, may fall into the error that a sphere is
a geometric figure in which one diameter is particularly distinguished
from the rest.
Footnote 19:
Einzelne stelle der Materie.
Footnote 20:
Vide Note.
Footnote 21:
_Vide_ note 13.
Footnote 22:
_Vide_ note 14.
Public-domain text, read in full here on John Shaqi.
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