Page:The principle of relativity (1920).djvu/90

 with six coefficients [function]_{2 3}-[function]_{3 4}. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.

x_{1}, x_{2}, x_{3}; y_{1}, y_{2}, y_{3}; [function]_{2 3}, [function]_{3 1}, [function]_{1 2}; [function]_{1 4}, [function]_{2 4}, [function]_{3 4} and the constants x_{4} and y_{4}, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject (x_{1}, x_{2}, x_{3}, x_{4}) and (y_{1}, y_{2}, y_{3}, y_{4}) simultaneously to the Lorentz transformation (21), the combination (23) is changed to.

(24) [function]_{2 3}´(x_{2}´y_{3}´ - x_{3}´y_{2}) + [function]_{3 1}(x_{3}´y_{1}´ - x_{1}´y_{3}) + [function]_{1 2}(x_{1}´y_{2}´ - x_{2}´y_{1}´) + [function]_{1 4}´(x_{1}´y_{4}´) - x_{4}´y_{1}´) + [function]_{2 4}´(x_{2}´y_{4}´ - x_{4}´y_{2}´) + [function]_{3 4}´(x_{3}´y_{4}´ - x_{4}´y_{3}´),

where the coefficients [function]_{2 3}´, [function]_{3 1}´, [function]_{1 2}´, [function]_{1 4}´, [function]_{2 4}´, [function]_{3 4}´, depend solely on ([function]_{2 3} [function]_{2 4}) and the coefficients a_{11} a_{44}.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes [function]_{2 3}, [function]_{3 1} [function]_{3 4}, with the condition that when subjected to a Lorentz transformation, it is changed to a new system [function]_{2 3}´ f_{3 4},  which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I)-(iv),—which are the fundamental equations for Äther.

If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time (pu_{x}, pu_{y}, pu_{z}, iρ) (convection-current, and charge density ρi) is transformed as a space time vector of the 1st kind, further (m_{x}, m_{y}, m_{z}, -ie_{x}, -ie_{y}, -ie_{z}) (magnetic force, and electric induction × (-i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.