Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_ _i.e._ We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36). The determinant of _f_ is therefore the square of the combination, by Det^{½}_f_ we shall denote the expression Det^{½}_f_ = _f₃₂_ _f₁₄_ _f₁₃_ _f₂₄_ + _f₂₁_ _f₃₄_· 4⁰. A linear transformation _x__{_h_} = α_{_h_1} _x₁′_ + α_{_h_2} _x₂_′ + α_{_h_3} _x₃′_ + α_{_h_4} _x₄′_ (_h_ = 1,2,3, which is accomplished by the matrix A = | α₁₁, α₁₂, α₁₃, α₁₄ | | | | α₂₁, α₂₂, α₂₃, α₂₄ | | | | α₃₁, α₃₂, α₃₃, α₃₄ | | | | α₄₁, α₄₂, α₄₃, α₄₄ | will be denoted as the transformation A. By the transformation A, the expression _x²₁_ + _x²₂_ + _x²₃_ + _x²₄_ is changed into the quadratic for _m_ ∑ α_{_hk_} _x__{_h_}′ _x__{_k_}′, where α_{_hk_} = α_{1_k_} α_{1_k_} + α_{2_h_} α_{2_k_} + α_{3_h_} α_{3_k_} + α_{4_h_} α_{4_k_} are the members of a 4 × 4 series matrix which is the product of Ā A, the transposed matrix of A into A. If by the transformation, the expression is changed to _x′₁²_ + _x₂′_^2 + _x₃′_^2 + _x′₄²_, we must have Ā A = 1. A has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that (Det A)² = 1, or Det A = ± 1. From the condition (39) we obtain A⁻¹ = Ā, _i.e._ the reciprocal matrix of A is equivalent to the transposed matrix of A. For A as Lorentz transformation, we have further Det A = +1, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and _a₄₄_ > 0. 5⁰. A space time vector of the first kind[21] which s represented by the 1 × 4 series matrix, (41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_| is to be replaced by _s_A in case of a Lorentz transformation A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_| A; A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_ shall be represented by the alternating matrix (42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ | |_f₂₁_ 0 _f_₂₃ _f₂₄_ | |_f_₃₁ _f₃₂_ 0 _f₃₄_ | |_f_₄₁ _f_₄₂ _f_₄₃ 0 |