Page:The principle of relativity (1920).djvu/123

 and from 83),

T_{x} = Ω_{1}, T_{y} = Ω_{2}, T_{z} = Ω_{3}

X_{t} = εμΩ_{1}, Y_{t} = εμΩ_{2}, Z_{t} = εμΩ_{3}

Now by means of a rotation of the space co-ordinate system round the null-point, we can make,

Z_{y} = Y_{z} = 0, X_{z} = Z_{x} = 0, X_{x} = X_{y} = 0,

According to 71), we have

(88) X_{x} + Y_{y} + Z_{z} + T_{t} = 0,

and according to 83), T_{t} > 0. In special cases, where Ω vanishes it follows from 81) that

X_{x}^2 = Y_{y}^2 = Z_{z}^2 = T_{t}^2, = (Det^{1/4} S)^2,

and if T, and one of the three magnitudes X_{x}, Y_{y}, Z_{z} are = ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let Ω [/=] 0, then we have in particular from 80)

T_{z} X_{t} = 0, T_{z} Y_{t} = 0, Z_{z} T_{z} + T_{z} T_{t} = O,

and if Ω_{1} = 0, Ω_{2} = 0, Z_{z} = -T_{t} It follows from (81), (see also 83) that

X_{x} = -Y_{y} = ±Det^{1/4} S,

and -Z_{z} = T_{t} = [sqrt](Det^{1/2} S + εμΩ_{3}^2) > Det^{1/4}S.—

The space-time vector of the first kind

(89) K = lor S,

is of very great importance for which we now want to demonstrate a very important transformation

According to 78), S = L + [function]F, and it follows that

lor S = lor L + lor [function]F.