Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/44

stands for what is called the vector product of the two vectors, namely the vector

$({{Y_{\alpha }}^{\prime }}{Z_{\alpha }}-{{Z_{\alpha }}^{\prime }}{Y_{\alpha }},{{Z_{\alpha }}^{\prime }}{X_{\alpha }}-{{X_{\alpha }}^{\prime }}{Z_{\alpha }},{{X_{\alpha }}^{\prime }}{Y_{\alpha }}-{{Y_{\alpha }}^{\prime }}{X_{\alpha }})$

It is evident that ${curl_{\alpha }}(X_{\alpha },Y_{\alpha },Z_{\alpha })$ can be expressed in the symbolic form

${\bigg [}{\bigg (}{\frac {\partial }{\partial x_{\alpha }}},{\frac {\partial }{\partial y_{\alpha }}},{\frac {\partial }{\partial z_{\alpha }}}{\bigg )}.(X_{\alpha },Y_{\alpha },Z_{\alpha }){\bigg ]}$

The vector equation

$(X_{\alpha },Y_{\alpha },Z_{\alpha })=({X_{\alpha }}^{\prime },{Y_{\alpha }}^{\prime },{Z_{\alpha }}^{\prime })$

is an abbreviation of the three equations

$X_{\alpha }={X_{\alpha }}^{\prime },Y_{\alpha }={Y_{\alpha }}^{\prime },Z_{\alpha }={Z_{\alpha }}^{\prime }$

Let $(F_{\alpha },G_{\alpha },H_{\alpha })$ be the electric force at $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$, and let $(L_{\alpha },M_{\alpha },N_{\alpha })$ be the magnetic force at the same point and time. Also let $\rho _{\alpha }$ a be the volume density of the electric charge and $(u_{\alpha },v_{\alpha },w_{\alpha })$ its velocity; and let $(P_{\alpha },Q_{\alpha },R_{\alpha })$ be the ponderomotive force: all equally at $(x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })$. Finally let $c$ be the velocity of light in vacuo.

Then Lorentz’s form of Maxwell’s equations is

(1) ${{\text{div}}_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=\rho _{\alpha }$

(2) ${{\text{div}}_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=0$

(3) ${curl_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=-{\frac {1}{c}}{\bigg \{}{\frac {\partial }{\partial t_{\alpha }}}(F_{\alpha },G_{\alpha },H_{\alpha })+{\rho _{\alpha }}(u_{\alpha },v_{\alpha },w_{\alpha }){\bigg \}}$

(4) ${curl_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=-{\frac {1}{c}}{\frac {\partial }{\partial t_{\alpha }}}(L_{\alpha },M_{\alpha },N_{\alpha })$

(5) $(P_{\alpha },Q_{\alpha },R_{\alpha })=(F_{\alpha },G_{\alpha },H_{\alpha })+{\frac {1}{c}}[(u_{\alpha },v_{\alpha },w_{\alpha }).(L_{\alpha },M_{\alpha },N_{\alpha })]$

It will be noted that each of the vector equations (3), (4), (5) stands for three ordinary equations, so that there are eleven equations in the five formulae.