Page:The principle of relativity (1920).djvu/81

 Therefore

[function]_{3 2} = -[function]_{2 3}, [function]_{1 3} = -[function]_{3 1}, [function]_{2 1} = -[function]_{1 2} [function]_{4 1} = -[function]_{1 4}, [function]_{4 4} = -[function]_{2 4}, [function]_{4 3} = -[function]_{3 4}

Then the three equations comprised in (i), and the equation (ii) multiplied by i becomes


 * δ[function]_{1 2}/δx_{2} + δ[function]_{1 3}/δx_{3} + δ[function]_{1 4}/δx_{4} = ρ_{1} |
 * δ[function]_{2 1}/δx_{1} + δ[function]_{2 3}/δx_{3} × δ[function]_{2 4}/δx_{4} = ρ_{2} |
 * |(A)
 * δ[function]_{3 1}/δx_{1} × δ[function]_{3 2}/δx_{2} + δ[function]_{3 4}/δx_{4} = ρ_{3} |
 * δ[function]_{4 1}/δx_{1} + δ[function]_{4 2}/δx_{2} + δ[function]_{4 3}/δx_{3} = ρ_{4} | x

On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by (i) becomes


 * δ[function]_{3 4}/δx_{2} + δ[function]_{4 2}/δx_{3} + δ[function]_{2 3}/δx_{4} = 0 |
 * δ[function]_{4 3}/δx_{1} + δ[function]_{1 4}/δx_{3} + δ[function]_{3 1}/δx_{4} = 0 |
 * | (B)
 * δ[function]_{2 4}/δx_{1} + δ[function]_{4 1}/δx_{2} + δ[function]_{1 2}/δx_{4} = 0 |
 * δ[function]_{3 2}/δx_{1} + δ[function]_{1 3}/δx_{2} + δ[function]_{2 1}/δx_{3} = 0 | x

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices. (1, 2, 3, 4).

§ 3.

It is well-known that by writing the equations i) to iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the