Then if S is, as in (62), a space-time matrix of the II kind, by lor S′
will be understood the 1 × 4 series matrix
| K₁ K₂ K₃ K₄ |
where K_{_k_} = ∂S_{1_k_}/∂_x₁_ + ∂S_{2_k_}/∂_x₂_ + ∂S_{3_k_}/∂_x₃_ +
∂S_{4_h_}/∂_x₄_.
When by a Lorentz transformation A, a new reference system (_x′₁_ _x′₂_
_x′₃_ _x₄_) is introduced, we can use the operator
lor′ = | ∂/∂_x₁′_ ∂/∂_x₂′_ ∂/∂_x₃′_ ∂/∂_x₄′_ |
Then S is transformed to S′= Ā S A = | S′_{_hk_} |, so by lor 'S′ is
meant the 1 × 4 series matrix, whose element are
K’_{_k_} = ∂S′_{1_k_}/∂_x₁′_ + ∂S′_{2_k_}/∂_x₂′_
+ ∂S′_{3_k_}/∂_x₃′_ + ∂S′_{4_k_}/∂_x₄′_.
Now for the differentiation of any function of (_x_ _y_ _z_ _t_) we have
the rule ∂/∂_x__{_k_}′ = ∂/∂_x₁_ ∂_x₁_/∂_x__{_k_}′ + ∂/∂_x₂_
∂_x₂_/∂_x__{_k_}′ + ∂/∂_x₃_ ∂_x₃_/∂_x__{_k_}′ + ∂/∂_x₄_
∂_x₄_/∂_x__{_k_}′ = ∂/∂_x₁_ _a__{1_k_} + ∂/∂_x₂_ _a__{2_k_} + ∂/∂_x₃_
_a__{3_k_} + ∂/∂_x₄_ _a__{4_k_}.
so that, we have symbolically lor′ = lor A.