Page:Lorentz Simplified1899.djvu/4



{{MathForm2|(III{{sub|c}})|$$\left.\begin{array}{l} \frac{\partial\mathfrak{H}'_{z}}{\partial y'}-\frac{\partial\mathfrak{H}'_{y}}{\partial z'}=4\pi k^{2}\varrho\mathfrak{v}_{x}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{x}}{\partial t'}\\ \\\frac{\partial\mathfrak{H}'_{x}}{\partial z'}-\frac{\partial\mathfrak{H}'_{z}}{\partial x'}=4\pi k\varrho\mathfrak{v}_{y}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{y}}{\partial t'}\\ \\\frac{\partial\mathfrak{H}'_{y}}{\partial x'}-\frac{\partial\mathfrak{H}'_{x}}{\partial y'}=4\pi k\varrho\mathfrak{v}_{z}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{z}}{\partial t'}\end{array}\right\} $$}}

{{MathForm2|(IV{{sub|c}})|$$\left.\begin{array}{l} \frac{\partial\mathfrak{F}'_{z}}{\partial y'}-\frac{\partial\mathfrak{F}'_{y}}{\partial z'}=-\frac{\partial\mathfrak{H}'_{x}}{\partial t'}\\ \\\frac{\partial\mathfrak{F}'_{x}}{\partial z'}-\frac{\partial\mathfrak{F}'_{z}}{\partial x'}=-\frac{\partial\mathfrak{H}'_{y}}{\partial t'}\\ \\\frac{\partial\mathfrak{F}'_{y}}{\partial x'}-\frac{\partial\mathfrak{F}'_{x}}{\partial y'}=-\frac{\partial\mathfrak{H}'_{z}}{\partial t'}\end{array}\right\} $$}}

{{MathForm2|(V{{sub|c}})|$$\left.\begin{array}{l} \mathfrak{E}_{x}=\mathfrak{F}'_{x}+k\frac{\mathfrak{p}_{x}}{V^{2}}(\mathfrak{v}_{y}\mathfrak{F}'_{y}+\mathfrak{v}_{z}\mathfrak{F}'_{z})+(\mathfrak{v}_{y}\mathfrak{H}'_{z}-\mathfrak{v}_{z}\mathfrak{H}'_{y})\\ \\\mathfrak{E}_{y}=\frac{1}{k}\mathfrak{F}'_{y}-k\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\mathfrak{F}'_{y}+(\frac{1}{k}\mathfrak{v}_{z}\mathfrak{H}'_{x}-\mathfrak{v}_{x}\mathfrak{H}'_{z})\\ \\\mathfrak{E}_{z}=\frac{1}{k}\mathfrak{F}'_{z}-k\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\mathfrak{F}'_{z}+(\mathfrak{v}_{x}\mathfrak{H}'_{y}-\frac{1}{k}\mathfrak{v}_{y}\mathfrak{H}'_{x})\end{array}\right\}$$.}}

Putting $$\mathfrak{v}=0$$ in the three last equations we see that

$$\mathfrak{F}'_{x},\ \frac{1}{k}\mathfrak{F}'_{y},\ \frac{1}{k}\mathfrak{F}'_{z}$$

are the components of the electric force that would act on a particle at rest.

§ 5. We shall begin with an application of the equations to