# Schwere, Elektricität und Magnetismus:378

Bernhard Riemann: Schwere, Elektricität und Magnetismus :
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Now since $\sigma =\alpha +\beta \,$ we may form a triangle in space, the sides of which shall be $\beta \,$, $\gamma \,$, and $\sigma \,$. Projecting this on a plane perpendicular to $\alpha \,$, we obtain a triangle having the sides $\beta ''\,$, $\gamma ''\,$, and $\sigma ''\,$, which affords the relation $\sigma ''=\beta ''+\gamma ''\,$. If we pass planes perpendicular to$\alpha \,$ through the vertices of the first triangle, they will give on a line parallel to $\alpha \,$ segments equal to $\beta ',\,\gamma ',\,\sigma '$. Thus we obtain the relation $\sigma '=\beta '+\gamma '\,$. Therefore $\alpha \cdot \sigma '=\alpha \cdot \beta '+\alpha \cdot \gamma '\,$, since all the cosines involved in these products are equal to unity. Moreover, if $\alpha \,$ is a unit vector, we shall evidently have $\alpha \times \sigma ''=\alpha \times \beta ''+\alpha \times \gamma ''\,$, since the effect of the skew multiplication by $\alpha \,$ upon vectors in a plane perpendicular to $\alpha \,$ is simply to rotate them all 90° in that plane. But any case may be reduced to this by dividing both sides of the equation to be proved by the magnitude of $\alpha \,$. The propositions are therefore proved.
20. Hence,
* $\lbrack \alpha +\beta \rbrack \cdot \gamma =\alpha \cdot \gamma +\beta \cdot \gamma ,\quad \lbrack \alpha +\beta \rbrack \times \gamma =\alpha \times \gamma +\beta \times \gamma ,\,$
* $\lbrack \alpha +\beta \rbrack \cdot \lbrack \gamma +\delta \rbrack =\alpha \cdot \gamma +\alpha \cdot \delta +\beta \cdot \gamma +\beta \cdot \delta ,\,$
* $\lbrack \alpha +\beta \rbrack \times \lbrack \gamma +\delta \rbrack =\alpha \times \gamma +\alpha \times \delta +\beta \times \gamma +\beta \times \delta ;\,$
and, in general, direct and skew products of sums of vectors may bo expanded precisely as the products of sums in algebra, except that in skew products the order of the factors must not be changed without compensation in the sign of the term. If any of the terms in the factors have negative signs, the signs of the expanded product (when there is no change in the order of the factors), will be determined by the same rules as in algebra. It is on account of this analogy with algebraic products that these functions of vectors are called products and that other terms relating to multiplication are applied to them.
21. Numerical calculation of direct and skew products.—The properties demonstrated in the last two paragraphs (which may be briefly expressed by saying that the operations of direct and skew multiplication are distributive) afford the rule for the numerical calculation of a direct product, or of the components of a skew product, when the rectangular components of the factors are given numerically. In fact,
| if | $\alpha =xi+yj+zk,\,$ and $\beta =x'i+y'j+z'k;\,$ |
* $\alpha \cdot \beta =xx'+yy'+zz',$
| and | $\alpha \times \beta =(yz'-zy')i+(zx'-xz')j+(xy'+yx')k.$ |
22. Representation of the area of a parallelogram by a skew product.—It will be easily seen that $\alpha \times \beta \,$ represents in magnitude the area of the parallelogram of which $\alpha \,$ and $\beta \,$ (supposed drawn from a common origin) are the sides, and that it represents in direction the normal to the plane of the parallel-