Page:Elektrische und Optische Erscheinungen (Lorentz) 048.jpg

Investigation of oscillations excited by oscillating ions.

General formulas.

§ 30. Once the motion of the ions is given, known functions of x, y, z and t appear on the right-hand side of equations (A) and (B) (§ 21); with respect to the last variable, these are periodic functions if the ions carry out oscillations with constant amplitude and a common oscillation interval T. It is easy to see, that in this case the equations are satisfied by values of ${\mathfrak {d}}_{x}$, ${\mathfrak {d}}_{y}$, ${\mathfrak {d}}_{z}$, ${\mathfrak {H}}_{x}$, ${\mathfrak {H}}_{y}$, ${\mathfrak {H}}_{z}$, which also have the period T. Therefore, the important and almost self-evident theorem is given:

If ion oscillations of period T take place in a light source, then ${\mathfrak {d}}$ and ${\mathfrak {H}}$ indicate the same periodicity at each point that shares the translation of the source.

The resolution of the equations leads to quite complicated expressions. For simplicity, it is advisable to calculate the components of the vector ${\mathfrak {H}}'$ (§ 20) at first.

According to (VI_b)

${\mathfrak {H}}'_{x}={\mathfrak {H}}_{x}-4\pi \left({\mathfrak {p}}_{y}{\mathfrak {d}}_{z}-{\mathfrak {p}}_{z}{\mathfrak {d}}_{y}\right).$

Accordingly, we want to multiply the second and third of equations (A) by $4\pi {\mathfrak {p}}_{z}$ and $-4\pi {\mathfrak {p}}_{y}$ respectively, and then add them to the first of equations (B). We obtain in this way, under consideration of the importance of $\left({\tfrac {\partial }{\partial t}}\right)_{1}$ (§ 19),