# Schwere, Elektricität und Magnetismus:380

Bernhard Riemann: Schwere, Elektricität und Magnetismus :
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two components, $\beta '\!$ and $\beta ''\!$, respectively parallel and perpendicular to $\alpha \!$. Then
* $\beta =\beta '+\beta '',\,$
* $(\alpha \cdot \beta )\alpha =(\alpha \cdot \beta ')\alpha =\alpha \cdot \alpha )\beta ',\!$
* $\alpha \times \lbrack \alpha \times \beta \rbrack =\alpha \times \lbrack \alpha \times \beta ''\rbrack =-(\alpha \cdot \alpha )\beta ''.\!$
| Hence, | $\alpha \times \lbrack \alpha \times \beta \rbrack =(\alpha \cdot \beta )\alpha -(\alpha \cdot \alpha )\beta .\,$ |
27. General relation of the vector products of three factors.—In the triple product $\alpha \times \lbrack \beta \times \gamma \rbrack$ we may set
* $\alpha =l\beta +m\gamma +n\beta \times \gamma ,\,$
unless $\beta \!$ and $\gamma \!$ have the same direction. Then
* $\alpha \times \lbrack \beta \times \gamma \rbrack =l\beta \times \lbrack \beta \times \gamma \rbrack +m\gamma \times \lbrack \beta \times \gamma \rbrack \,$
* $=l(\beta \cdot \gamma )\beta -l(\beta \cdot \beta )\gamma -m(\gamma \cdot \beta )\gamma +m(\gamma \cdot \gamma )\beta \,$
* $=(l\beta \cdot \gamma +m\gamma \cdot \gamma )\beta -(l\beta \cdot \beta +m\gamma \cdot \beta )\gamma .\,$
| But | $l\beta \cdot \gamma +m\gamma \cdot \gamma =\alpha \cdot \gamma \,$ and $l\beta \cdot \beta +m\gamma \cdot \beta =\alpha \cdot \beta .\,$ |
| Therefore | $\alpha \times \lbrack \beta \times \gamma \rbrack =(\alpha \cdot \gamma )\beta -(\alpha \cdot \beta )\gamma ,\,$ |
which is evidently true, when $\beta \!$ and $\gamma \!$ have the same directions. It may also be written
* $\lbrack \beta \times \gamma \rbrack \times \alpha =\beta (\gamma \cdot \alpha )-\gamma (\beta \cdot \alpha ).\,$
28. This principle may be used in the transformation of more complex products. It will be observed that its application will always simultaneously eliminate, or introduce, two signs of skew multiplication.
The student will easily prove the following identical equations, which, although of considerable importance, are here given principally as exercises in the application of the preceding formulae.
29. $\alpha \times \lbrack \beta \times \gamma \rbrack +\beta \times \lbrack \gamma \times \alpha \rbrack +\gamma \times \lbrack \alpha \times \beta \rbrack =0.\,$
30. $\lbrack \alpha \times \beta \rbrack \cdot \lbrack \gamma \times \delta \rbrack =(\alpha \cdot \gamma )(\beta \cdot \delta )-(\alpha \cdot \delta )(\beta \cdot \gamma ).\,$
31. $\lbrack \alpha \times \beta \rbrack \times \lbrack \gamma \times \delta \rbrack =(\alpha \cdot \gamma \times \delta )\beta -(\beta \cdot \gamma \times \delta )\alpha =(\alpha \cdot \beta \times \delta )\gamma -(\alpha \cdot \beta \times \gamma )\delta .\,$
32. $\alpha \times \lbrack \beta \times \lbrack \gamma \times \delta \rbrack \rbrack =(\alpha \cdot \gamma \times \delta )\beta -(\alpha \cdot \beta )\gamma \times \delta =(\beta \cdot \delta )\alpha \times \gamma -(\beta \cdot \gamma )\alpha \times \delta .\,$
33. $\lbrack \alpha \times \beta \rbrack \times \lbrack \gamma \times \delta \rbrack \times \lbrack \varepsilon \times \zeta \rbrack =(\alpha \cdot \beta \times \delta )(\gamma \cdot \varepsilon \times \zeta )-(\alpha \cdot \beta \times \gamma )(\delta \cdot \varepsilon \times \zeta )\,$
$=-(\alpha \cdot \beta \times \zeta )(\varepsilon \cdot \gamma \times \delta )+(\alpha \cdot \beta \times \varepsilon )(\zeta \cdot \gamma \times \delta )\,$
$=(\gamma \cdot \delta \times \alpha )(\beta \cdot \varepsilon \times \zeta )+(\gamma \cdot \delta \times \beta )(\alpha \cdot \varepsilon \times \zeta ).\,$
34. $\lbrack \alpha \times \beta \rbrack \cdot \lbrack \beta \times \gamma \rbrack \times \lbrack \gamma \times \alpha \rbrack =(\alpha \cdot \beta \times \gamma )^{2}.\,$