Page:The principle of relativity (1920).djvu/124

 The symbol 'lor' denotes a differential process which in lor fF, operates on the one hand upon the components of f, on the other hand also upon the components of F. Accordingly lor fF can be expressed as the sum of two parts. The first part is the product of the matrices (lor f) F, lor f being regarded as a 1 × 4 series matrix. The second part is that part of lor fF, in which the diffentiations operate upon the components of F alone. From 78) we obtain

fF = -F^*f^* - 2L;

hence the second part of lor fF = -(lor F^*)f^* + the part of -2 lor L, in which the differentiations operate upon the components of F alone. We thus obtain

lor S = (lor f)F - (lor F^*)f^* + N,

where N is the vector with the components

N_{h} = 1/2([part]f_{23}/[part]x_{h} F_{23} + [part]f_{31}/[part]x_{h} F_{31} + [part]f_{12}/[part]x_{h} F_{12} + [part]f_{14}/[part]x_{h} F_{14} + [part]f_{24}/[part]x_{h} F_{21} + [part]f_{34}/[part]x_{h} F_{34} - [part]F_{23}/[part]x_{h} f_{23} - [part]F_{31}/[part]x_{h} f_{31} - [part]F_{12}/[part]x_{h} f_{12} - [part]F_{14}/[part]x_{h} f_{14} - [part]F_{24}/[part]x_{h} f_{24} - [part]F_{34}/[part]x_{h} f_{3-}),

(h = 1, 2, 3, 4)

By using the fundamental relations A) and B), 90) is transformed into the fundamental relation

(91) lor S = -sF + N.

In the limitting case ε = 1, μ = 1, f = F, N vanishes identically.