Page:The principle of relativity (1920).djvu/119

 are of much importance. Let us write.

(70) [function]F =| S_{11} - L S_{12}  S_{13}  S_{14} |

| S_{21} S_{22} - L  S_{23}  S_{24} |

| S_{31} S_{32}  S_{33} - L  S_{34} |

| S_{41} S_{42}  S_{43}  S_{44} - L |

Then (71) S_{11} + S_{22} + S_{33} + S_{44} = 0.

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

(72) L = 1/2([function]_{23} F_{23} + [function]_{31}F_{31} + [function]_{12} + F_{12} + [function]_{14} F_{14} + [function]_{24} F_{24} + [function]_{34} F_{34})

Then we shall have

(73) S_{11} = 1/2([function]_{23} F_{23} + [function]_{34} F_{34} + [function]_{42} F_{42} - [function]_{12} F_{12} - [function]_{13} F_{13} [function]_{14} F_{14})

S_{12} = [function]_{13} F_{32} + [function]_{14} F_{42} etc

In order to express in a real form, we write

(74) S = | S_{11} S_{12} S_{13} S_{14} |

| S_{21} S_{22} S_{23} S_{24} |

| S_{31} S_{32} S_{33} S_{34} |

| S_{41} S_{42} S_{43} S_{44} |

= | X_{x} Y_{x} Z_{x} -iT_{x} |

| X_{y} Y_{y} Z_{y} -iT_{y} |

| X_{z} Y_{z} Z_{z} -iT_{z} |

| -iX_{t} -iY_{t} -iZ_{t} T_{t} |

Now X_{x} = 1/2[m_{x}M_{x} - m_{y}M_{y} - m_{z}M_{z} + e_{x}E_{x} - e_{y}E_{y} - e_{z}E_{z}]