Page:The principle of relativity (1920).djvu/257

 h_{x} h_{y} h_{z}), in accordance with the notation used in Lorentz's Theory of Electrons.

We have therefore

ρ_{0} (phi]_{1}, φ_{2}, φ_{3}) = ρ (d + 1/c [v·h]),

i.e., ρ_{0} (phi]_{1}, φ_{2}, φ_{3}) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.

The fourth component φ_{4} when multiplied by ρ_{0} represents i-times the rate at which work is done by the moving electron, for ρ_{0} φ_{4} = iρ [v_{x}d_{x} + v_{y}d_{y} + v_{z}d_{z}] = v_{x} ρ_{0}φ_{1} + v_{y} ρ_{0}φ_{2} + v_{z} ρ_{0}φ_{3}. -[sqrt](_{-1}) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906.

The four-vector ψ =iωF^* has a similar relation to the force acting on a moving magnetic pole.

[M. N. S.]

Note 17. Operator "Lor"  (§ 12, p. 41).

The operation | [part]/[part]x_{1} [part]/[part]x_{2} [part]/[part]x_{3} [part]/[part]x_{4} | which plays in four-dimensional mechanics a rôle similar to that of the operator (i[part]/[part]x, + j[part]/[part]y, + k[part]/[part]z = [nabla] in three-dimensional geometry has been called by Minkowski 'Lorentz-Operation' or shortly  'lor'  in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol  to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will