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It is obvious that linear one-dimensional extensions can be called "straight lines", also it will be clear what is to be understood by a "prism" (or "cylinder"). This latter is bounded by two mutually parallel linear three-dimensional extensions $\sigma _{1}$ and $\sigma _{s}$ and by a lateral surface which may be extended indefinitely to both sides and in which mutually parallel straight lines ("generating lines") can be drawn.

We need not dwell upon the elementary properties of the prism.

§ 17. A vector may now be represented by a straight line of finite length; the quantities $X_{1},\dots X_{4}$, which have been introduced in § 10, are the changes of the coordinates caused by a displacement along that line. The magnitude of the vector, expressed in natural units, will be denoted by $S$. It is given by a formula similar to (1), viz. by

$S^{2}=\sum (ab)g_{ab}X_{a}X_{b}$

(11)

A vector may be regarded as being the same everywhere in the field-figure, if $X_{1},\dots X_{4}$ have constant values. In the same way a rotation $\mathrm {R}$ (§ 11) may be said to be the same everywhere, if it can be represented by two vectors of this kind.

If from a point $P$ two vectors $PQ$ and $PR$ issue, denoted by $X'_{1},\dots X'_{4}$, $S'$ and $X''_{1},\dots X''_{4}$, $S''$ resp., the angle between them (comp. (5)) is defined by

$S'S''\cos(S',S'')=\sum (ab)g_{ab}X'_{a}X''_{b}$

(12)

We remark here that $X'_{a},\ X''_{b}$ are real, positive or negative quantities and that $S'$ and $S''$ are expressed in the way indicated in § 5 ("absolute" values). It is to be understood that $S$ does not change when the signs of $X_{1},\dots X_{4}$ are reversed at the same time.

If $S'''$ is the value of the vector $RQ$ and if the angle between this vector and $RP$ is denoted by ($S'',S'''$), it follows further from (11) and (12) that

$S''=S'\cos(S',S'')+S'''\cos(S'',S''')$

In the special case of a right angle $R$ we have

$S''=S'\cos(S',S'')$

an equation expressing the connexion between a vector $PQ$ and its "projection" on a line $PR$. The angle ($S',S''$) is the angle between the vector and its projection, both reckoned from the same point $P$.

§ 18. Let us now return to the prism $R$ mentioned in § 16. From a point $A_{2}$ of the boundary of the "upper face"$\sigma _{2}$, we can