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useful for finding invariants.^[1] Various invariants are calculated with a view of obtaining the general form of the equations of motion. Abraham's expression of the reaction of radiation on a moving electron is shown to be an invariant for the group of spherical wave transformations, provided it is multiplied by a certain factor.

2. The Integral Equations of the Theory of Electrons.

Let the electromagnetic units be those adopted by Lorentz, and let the units of time and length be so chosen that the velocity of radiation is unity.

Let curl H denote the vector whose components are

${\frac {\partial H_{z}}{\partial y}}-{\frac {\partial H_{y}}{\partial z}},\dots ,$

when the axes of x, y, z are right-handed and the vector whose components are

${\frac {\partial H_{y}}{\partial z}}-{\frac {\partial H_{z}}{\partial y}},\dots ,$

when the axes are left-handed. Then the fundamental equations^[2] of the theory of electrons may be written

$\left.{\begin{array}{l}{\mathsf {curl}}\ H={\frac {\partial E}{\partial t}}+\rho w\\\\{\mathsf {curl}}\ E=-{\frac {\partial H}{\partial t}}\\\\{\mathsf {div}}\ E=\rho \\\\{\mathsf {div}}\ H=0\end{array}}\right\},$

(I)

where E is the electric force, H the magnetic force, ρ the volume density of electricity, and ρw the convection current.

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1. They appear to be closely connected with two theorems used by Ricci and Levi Civita, "Méthodes de calcul différential absolu et leurs applications," Math. Ann., Bd. LIV (1901).
2. H. A. Lorentz, Ency. d. math. Wiss., Vol. 13, No. 5 ; and Vol. 14, No. 2. The Theory of Electrons, Leipzig (1909), p. 12.