169.

\sin\frac{\mathsf{a} \pi x}{l} \sin\frac{\mathsf{b} \pi y}{l} \cos\frac{\mathsf{c} \pi z}{l} (e } \cos 2\pi \nu t + e_{3}' \sin 2\pi \nu t), \ \mathsf{H {x} &= \sin\frac{\mathsf{a} \pi x}{l} \cos\frac{\mathsf{b} \pi y}{l} \cos\frac{\mathsf{c} \pi z}{l} (h ' \cos 2\pi \nu t), } \sin 2\pi \nu t - h_{1

$$\begin{gathered} {𝖧}_y\&=\cos \frac{𝖺\pi x}{l}\sin \frac{𝖻\pi y}{l}\cos \frac{𝖼\pi z}{l}(h_2\sin 2\pi \nu t-h_2^{\prime }\cos 2\pi \nu t), \\ {𝖧}_z\&=\cos \frac{𝖺\pi x}{l}\cos \frac{𝖻\pi y}{l}\sin \frac{𝖼\pi z}{l}(h_3\sin 2\pi \nu t-h_3^{\prime }\cos 2\pi \nu t), \end{gathered}$$

\Label[eqn]{(305)}\tag*{\upshape (305)}$ where $𝖺$, $𝖻$, $𝖼$ represent any three positive integral numbers. The boundary conditions in these expressions are satisfied by the fact that for the six bounding surfaces $x=0$, $x=l$, $y=0$, $y=l$, $z=0$, $z=l$ the tangential components of the electric field-strength $⋿$ vanish. Maxwell's equations of the field [eqn:(52)] (52) are also satisfied, as may be seen on substitution, provided there exist certain conditions between the constants which may be stated in a single proposition as follows: Let $a$ be a certain positive constant, then there exist between the nine quantities written in the following square: $$\begin{gathered} c*2>c{\displaystyle \frac{𝖺c}{2l\nu }}\&{\displaystyle \frac{𝖻c}{2l\nu }}\&{\displaystyle \frac{𝖼c}{2l\nu }} \\ \mathrm{\backslash Strut}{\displaystyle \frac{h_1}{a}}\&{\displaystyle \frac{h_2}{a}}\&{\displaystyle \frac{h_3}{a}} \\ \mathrm{\backslash Strut}{\displaystyle \frac{e_1}{a}}\&{\displaystyle \frac{e_2}{a}}\&{\displaystyle \frac{e_3}{a}} \end{gathered}$$ all the relations which are satisfied by the nine so-called "direction cosines" of two orthogonal right-handed coordinate systems, i.e., the cosines of the angles of any two axes of the systems.

Hence the sum of the squares of the terms of any horizontal or vertical row equals $1$, for example, $\frac{c^{2}}{4l^{2}} \nu^{2} (\mathsf{a}^{2} + \mathsf{b}^{2} + \mathsf{c}^{2}) = 1 \ h_{1}^{2} + h_{2}^{2}