§ 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 system of equations [A)] as well as of [B)], when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis, through an amount φ, keeping e, m fixed in space, and introduce new variables x₁′ x₂′ x₃′ x₄′ instead of x₁ x₂ x₃ x₄ where x′₁ = x₁ cos φ + x₂ sin φ, x′₂ = -x₁ sin φ + x₂ cos φ, x′₃ = x₃, x′₄ = x₄, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and f′_{1 2}, ... ... f′_{3 4}, where

f′₂₃ = f₂₃ cos φ + f₃₁ sin φ,

f′₃₁ = - f₂₃ sin φ + f₃₁ cos φ,

f′₁₂ = f₁₂,

f′₁₄ = f₁₄ cos φ + f₂₄ sin φ,

f′₂₄ = - f₁₄ sin φ + f₂₄ cos φ,

f′₃₄ = f₃₄_{3 4},

f′_{k h} = - f_{k h} (h l k = 1, 2, 3, 4).

then out of the equations (A) would follow a corresponding system of dashed equations (A´) composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the suffixes (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote by iψ a purely imaginary magnitude, and consider the substitution

x₁′ = x₁,

x₂′ = x₂,

x₃′ = x₃ cos iψ + x₄ sin iψ, (1)

x₄′´ = - x₃ sin iψ + x₄ cos iψ,

Putting

"(2)."

We shall have cos iψ = 1/√(1 - q²), sin iψ = iq/√(1 - q²) where -1 < q < 1, and √(1 - q²) is always to be taken with the positive sign.

Let us now write x′₁ = x′, x′₂ = y′, x′₃ = z′, x′₄ = it′ (3)

then the substitution 1) takes the form

x′ = x, y′ = y, z′ = (z - qt)/√(1 - q²), t′ = (-qz + t)/√(1 - q²), (4)

the coefficients being essentially real.

If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ by iψ, we at once perceive that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where

ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cos iψ + ρ₄ sin iψ,

ρ′₄ = - ρ₃ sin iψ + ρ₄ cos iψ),

and f′_{1 2} ... f′_{3 4}, where

f′_{4 1} = f_{4 1} cos iψ + f_{1 3} sin iψ,

f′_{1 3} = - f_{4 1} sin iψ + f_{1 3} cos iψ,

f′_{3 4} = f_{3 4},

f′_{3 2} = f_{3 2} cos iψ + f_{4 2} sin iψ,

f′_{4 2} = - f_{3 2} sin iψ + f_{4 2} cos iψ,

f′_{1 2} = f_{1 2}, f_{k h} = - f′_{k h},

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, and x´ y´ z´ t´ be taken as a new frame of reference, then we shall have

(5) ρ´ = ρ [(-qu_z + 1)/√(1 - q²)],

ρ´u_z´ = ρ[(u_z - q)/√(1 - q²)],

ρ´u_x´ = ρu_x,

ρ´u_y´ = ρu_y.

(6) e´_x´ = (e_x - qm_y)/(√(1 - q²)),

m´_r´ = (qe_x + m_y)/(√(1 - q²)),

e´_z´ = e_z.

(7) m´_x´ = (m_x - qe_y)/(√(1 - q²)),

e´_y´ = (qm_x + e_y)/(√(1 - q²)),

m´_z´ = m_z.

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

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