Schwere, Elektricität und Magnetismus:384

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

where $\alpha ,\,\beta ,\,\gamma ,\,\delta ,\,\lambda ,\,\mu ,$ and $\nu \,$ represent known vectors. Multiplying directly by $\beta \times \gamma ,\,$ by $\gamma \times \alpha ,\,$ and by $\alpha \times \beta ,\,$ we obtain

$\beta \cdot \gamma \times \delta =(\beta \cdot \gamma \times \alpha )(\lambda \cdot \rho ),\quad \gamma \cdot \alpha \times \delta =(\gamma \cdot \alpha \times \beta )(\mu \cdot \rho ),$ $\alpha \cdot \beta \times \delta =(\alpha \cdot \beta \times \gamma )(\nu \cdot \rho );\,$

or	$\alpha '\cdot \delta =\lambda \cdot \rho ,\quad \beta '\cdot \delta =\mu \cdot \rho ,\quad \gamma '\cdot \delta =\nu \cdot \rho ,\,$

where $\alpha ',\,\beta ',\,\gamma '$ are the reciprocals of $\alpha ,\,\beta ,\,\gamma$. Substituting these values in the identical equation

$\rho =\lambda '(\lambda \cdot rho)+\mu '(\mu \cdot \rho )+\nu '(\nu .\rho ),\,$

in which $\lambda ',\,\mu ',\,\nu '$ are the reciprocals of $\lambda ,\,\mu ,\,\nu ,$ (see No. 38,) we have

$\rho =\lambda '(\alpha '\cdot \delta )+\mu '(\beta '\cdot \delta )+\nu '(\gamma '\cdot \delta ),\,$

(2)

which is the solution required.

 It results from the principle stated in No. 35, that any vector equation of the first degree with respect to $\rho \,$ may be reduced to the form

$\delta =\alpha (\lambda \cdot \rho )+\beta (\mu \cdot \rho )+\gamma (\nu \cdot \rho )+a\rho +\varepsilon \times \rho .\,$

But	$a\rho =a\lambda '(\lambda \cdot \rho )+a\mu '(\mu \cdot \rho )+a\nu '(\nu \cdot \rho ),\,$

and	$\varepsilon \times \rho =\varepsilon \times \lambda '(\lambda \cdot \rho )+\varepsilon \times \mu '(\mu \cdot \rho )+\varepsilon \times \nu '(\nu \cdot \rho ),\,$

where $\lambda ',\,\mu ',\,\nu '$ represent, as before, the reciprocals of $\lambda ,\,\mu ,\,\nu$. By substitution of these values the equation is reduced to the form of equation (1), which may therefore be regarded as the most general form of a vector equation of the first degree with respect to $\rho \,$.

 41. Relations between two normal systems of unit vectors.—If $i,\,j,\,k$, and $i',\,j',\,k'$ are two normal systems of unit vectors, we have

${\begin{matrix}i'=(i\cdot i')i+(j\cdot i')j+(k\cdot i')k,\\j'=(i\cdot j')i+(j\cdot j')j+(k\cdot j')k,\\k'=(i\cdot k')i+(j\cdot k')j+(k\cdot k')k,\end{matrix}}{\Biggr \rbrace }$

(1)

and

${\begin{matrix}i=(i\cdot i')i'+(i\cdot j')j'+(i\cdot k')k',\\j=(j\cdot i')i'+(j\cdot j')j'+(j\cdot k')k',\\k=(k\cdot i')i'+(k\cdot j')j'+(k\cdot k')k'.\end{matrix}}{\Biggr \rbrace }$

(2)

(See equation 8 of No. 38.)

 The nine coefficients in these equations are evidently the cosines of the nine angles made by a vector of one system with a vector of the other system. The principal relations of these cosines are easily deduced. By direct multiplication of each of the preceding equations with itself, we obtain six equations of the type

$(i\cdot i')^{2}+(j\cdot i')^{2}+(k\cdot i')^{2}=1.\,$

(3)