Page:Elektrische und Optische Erscheinungen (Lorentz) 055.jpg

The direction of the wave normal.

§ 36. Now we shall examine the oscillations in such distances from the luminous molecules, which are considerably larger than the wavelength. It should be noted that in (39) and (40), ${\mathfrak {m}}_{x}$, ${\mathfrak {m}}_{y}$, ${\mathfrak {m}}_{z}$ are goniometric functions of

$t'-{\frac {r}{V}}$;

we namely want to write from now on r instead of $r_{0}$. The assumption made about the length of this line justifies to consider only the variability of the argument of any goniometric function for all differentiations with respect to x, y, z, but to consider as constant all factors such as ${\tfrac {1}{r}}$, or $\cos(r,x)$, by which these functions are multiplied.

For any of the magnitudes ${\mathfrak {H'}}_{x}$, ${\mathfrak {H'}}_{y}$, ${\mathfrak {H'}}_{z}$, ${\mathfrak {F}}_{x}$, ${\mathfrak {F}}_{y}$, ${\mathfrak {F}}_{z}$ - we will call them φ — it can therefore be found an expression of the form

$\phi =A\cos {\frac {2\pi }{T}}\left(t'-{\frac {r}{V}}+B\right){,}$

(41)

where A and B are indeed dependent on the length and the direction of line $Q_{0}P$ — $Q_{0}$ is the location of the molecule, and P is the considered external point —, but, if r were just big enough, it may be regarded as constant in a space that comprises many wavelengths. While x, y, z are the coordinates of P, we denote by ξ, η, ζ the coordinates of $Q_{0}$, and by $b_{x}$, $b_{y}$, $b_{z}$ the direction constants of the connection-line $Q_{0}P$. If we now replace in the formula (41) r by

$b_{x}(x-\xi )+b_{y}(y-\eta )+b_{z}(z-\zeta ){,}$

and $t'$ by the value (34), we obtain

$\phi =A\cos {\frac {2\pi }{T}}\left\{t-\left({\frac {b_{x}}{V}}+{\frac {{\mathfrak {p}}_{x}}{V^{2}}}\right)x-\left({\frac {b_{y}}{V}}+{\frac {{\mathfrak {p}}_{y}}{V^{2}}}\right)y-\right.$ $\left.-\left({\frac {b_{z}}{V}}+{\frac {{\mathfrak {p}}_{z}}{V^{2}}}\right)z+C\right\}{,}$

(42)

$C=B+{\frac {1}{V}}(b_{x}\xi +b_{y}\eta +b_{z}\zeta ).$