A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} | | S₂₁ S₂₂ S₂₃ S₂₄ | | S₃₁ S₃₂ S₃₃ S₃₄ | | S₄₁ S₄₂ S₄₃ S₄₄ | with the condition that in case of a Lorentz transformation it is to be replaced by ĀSA, may be called a space-time matrix of the II kind. We have examples of this in:— 1) the alternating matrix _f_, which corresponds to the space-time vector of the II kind,— 2) the product _f_F of two such matrices, for by a transformation A, it is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA, 3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} = ω_{_h_}Ω_{_k_}, lastly in a multiple L of the unit matrix of 4 × 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero. We shall have to do constantly with functions of the space-time point (_x_, _y_, _z_, _it_), and we may with advantage employ the 1 × 4 series matrix, formed of differential symbols,— | ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,| or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | For this matrix I shall use the shortened from “lor.”[25] 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. Therefore it follows that lor ′S′ = lor (A A⁻¹ SA) = (lor S)A. _i.e._, lor S behaves like a space-time vector of the first kind. If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements | ∂L/∂_x₁_ ∂L/∂_x₂_ ∂L/∂_x₃_ ∂L/∂_x₄_ | If _s_ is a space-time vector of the 1st kind, then lor _ṡ_ = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_. In case of a Lorentz transformation A, we have lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_. _i.e._, lor _s_ is an invariant in a Lorentz-transformation.