Page:The principle of relativity (1920).djvu/251

 The general six-vector [function] is composed from the vectors φ, φ^* in the following way:—

[function] = ρφ + ρ^* φ^*,

rho and rho^* denoting the contents of the pieces of mutually perpendicular planes composing [function]. The "conjugate Vector" [function]^* (or it may be called the complement of [function]) is obtained by interchanging ρ and ρ^*

We have,

[function]^* = ρ^*φ + ρφ^*

We can verify that

[function]_{y z}^* = [function]_{x l} etc. and [function]^2 = ρ^2 + ρ^*^2, ([function][function]^*) = 2ρρ^*. | [function] |^2 and ([function][function]^*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates. [M. N. S.] Note 12. Light-velocity as a maximum.

Page 23, and Electro-dynamics of Moving Bodies, p. 17.

Putting v = c - x, and w = c - λ, we get

V = (2c - (x + λ))/(1 + (c - x)(c - λ)/c^2) = (2c - (x + λ))/(c^2 + c^2 - (x + λ)c + xλ/c^2)

= c (2c - (x + λ))/(2c - (x + λ) + xλ/c)

Thus v < c, so long as | xλ | > 0.

Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W > c.

Let A and B be separated by distance l, and let velocity of a "signal" in the system S be W > c. Let the