Page:The principle of relativity (1920).djvu/137

 At every space-time point (as in § 13), let a space time matrix of the 2nd kind be known


 * S_{11} S_{12} S_{13} S_{14} | = | X_{x} Y_{x} Z_{x} -iT_{x} | (16) S = | S_{21} S_{22} S_{23} S_{24} | = | X_{y} Y_{y} Z_{y} -iT_{y} |  | S_{31} S_{32} S_{33} S_{34} | = | X_{z} Y_{z} Z_{z} -iT_{z} |  | S_{41} S_{42} S_{43} S_{44} | = | -iX_{t} -iY_{t} -iZ_{t} T_{t} |  where X_{n} Y_{x} X_{z}, T_{t} are real magnitudes.  For a virtual displacement in a space-time sichel (with the previously applied designation) the value of the integral  (17) W + δW = [integral][integral][integral][integral] ([sum]S_{h k} ([part](x_{k} + δx_{k}))/[part]x_{h} dx dy dz dt  extended over the whole range of the sichel, may be called the tensional work of the virtual displacement.  The sum which comes forth here, written in real magnitudes, is  X_{x} + Y_{y} + Z_{z} + T_{t} + X_{x} ([part]δx)/[part]x + X_{y} ([part]δx)/[part]y +  Z_{z} ([part]δz)/[part]z

- X_{t} ([part]δx/[part]t - + T_{x} ([part]δt)/[part]x +  T_{t} ([part]δt)/[part]t  we can now postulate the following minimum principle in mechanics.  If any space-time Sichel be bounded, then for each virtual displacement in the Sichel, the sum of the mass-works, and tension works shall always be an extremum for that process of the space-time line in the Sichel which actually occurs.

The meaning is, that for each virtual displacement,

([d(·δN + δW)]/dλ)_{λ = 0} = 0 (18)