The Principle of Relativity

by Albert Einstein; H. Minkowski

Then we have for these newly introduced vectors u´, e´, m´ (with components u_x´, u_y´, u_z´; e_x´, e_y´, e_z´; m_x´, m_y´, m_z´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e_x - qm_y, e_y + qm_x are components of the vector e + [vm], where v is a vector in the direction of the positive Z-axis, and | v | = q, and [vm] is the vector product of v and m; similarly -qe_x + m_y, m_x + qe_y are the components of the vector m - [ve].

The equations 6) and 7), as they stand in pairs, can be expressed as.

e′_x′ + im′_x′ = (e_x + im_x) cos iψ + (e_y + im_y) sin iψ,

e′_y′ + im′_y′ = - (e_x + im_x) sin iψ + (e_y + im_y) cos iψ,

e′_z′ + im′_z′ = e′_z + im_z.

If φ denotes any other real angle, we can form the following combinations:—

(e′_x′ + im′_x′) cos. φ + (e′_y″ + im′_y′) sin φ

= (e_x + im_x) cos. (φ + iψ) + (e_y + im_y) sin (φ + iψ),

= (e′_x′ + im′_x′) sin φ + (e′_y′ + im′_y′) cos. φ