It may be mentioned here that Lorentz's fundamental equations of the electron theory, viz.,
${\frac {\partial \gamma }{\partial y}}{\frac {\partial \beta }{\partial z}}=4\pi u,\ {\frac {\partial Q}{\partial z}}{\frac {\partial R}{\partial y}}={\frac {\partial \alpha }{\partial t}},$
$u={\frac {\partial f}{\partial t}}+\rho \xi ,\ P=4\pi c^{2}f,$
${\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}(\rho \xi )+{\frac {\partial }{\partial y}}(\rho \eta )+{\frac {\partial }{\partial z}}(\rho \zeta )=0$
${\frac {\partial f}{\partial x}}+{\frac {\partial g}{\partial y}}+{\frac {\partial h}{\partial z}}=\rho ,\ {\frac {\partial \alpha }{\partial x}}+{\frac {\partial \beta }{\partial y}}+{\frac {\partial \gamma }{\partial z}}=0,$


may be reduced to a symmetrical form by writing s = ict and putting
$\alpha +i{\frac {P}{c}}=p,\ \beta +i{\frac {Q}{c}}=q,\ \gamma +i{\frac {R}{c}}=r.$


The four mutually orthogonal vectors (A, B, C, D) whose components are respectively
(0, r, q, p), (r, 0, p, q), (q, p, 0, r), (p, q, r, 0)
satisfy the equations
$div\ A=4\pi \rho \xi ,\ div\ B=4\pi \rho \eta ,\ div\ C=4\pi \rho \zeta ,\ div\ D=4\pi \rho ic,$


where
$div\ M\equiv {\frac {\partial M_{1}}{\partial x}}+{\frac {\partial M_{2}}{\partial y}}+{\frac {\partial M_{3}}{\partial z}}+{\frac {\partial M_{4}}{\partial s}}$ if $M\equiv \left(M_{1},\ M_{2},\ M_{3},\ M_{4}\right)$
Again, if we put
$4\pi \rho \xi =X,\ 4\pi \rho \eta =Y,\ 4\pi \rho \eta =Z,\ 4\pi \rho ic=S,$


and introduce four new vectors $A_{1},\ B_{1},\ C_{1},\ D_{1}$ whose components are respectively
(S, Z, Y, X) (Z, S, X, Y) (Y, X, S, Z) (X, Y, Z, S),
we find
$div\ A_{1}=\nabla ^{2}p,\ div\ B_{1}=\nabla ^{2}q,\ div\ C_{1}=\nabla ^{2}r,\ div\ D_{1}=0,$


where
$\nabla ^{2}\phi \equiv {\frac {\partial ^{2}\phi }{\partial x^{2}}}+{\frac {\partial ^{2}\phi }{\partial y^{2}}}+{\frac {\partial ^{2}\phi }{\partial z^{2}}}+{\frac {\partial ^{2}\phi }{\partial s^{2}}}.$


Finally, if X, Y, Z, S can be derived from a potential function n so that
$X={\frac {\partial n}{\partial x}},\ Y={\frac {\partial n}{\partial y}},\ Z={\frac {\partial n}{\partial z}},\ S={\frac {\partial n}{\partial s}},$


we can form four mutually orthogonal vectors θ, Φ, ψ, χ whose components are respectively
(n, r, q, p). (r, n, p, q), (q, p, n, r), (p, q, r, n),
and the equations then take the simple form
${\begin{array}{ccccccccc}div\ \theta &=&div\ \phi &=&div\ \psi &=&div\ \chi &=&0,\\\\\nabla ^{2}p&=&\nabla ^{2}q&=&\nabla ^{2}r&=&\nabla ^{2}n&=&0,\\\\div\ A_{1}&=&div\ B_{1}&=&div\ C_{1}&=&div\ D_{1}&=&0.\end{array}}$

