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Some definitions and mathematical relations

§ 4. a. We want to say, that a rotation in a plane corresponds to a certain direction of the perpendicular, and namely it shall be the direction into that side, at which an observer must be located, so that for him the rotation is counter-clockwise.

b. The mutually perpendicular coordinate axes OX, OY, OZ are chosen by us, so that the direction of OZ corresponds to a rotation around a right angle of OX to OY.

c. A space, a surface and a line we denote by the letters $\tau$, $\sigma$ and $s$ throughout, and infinitely small parts by $d\ \tau$, $d\ \sigma$ and $d\ s$.

The perpendicular to a surface will by sketched by n, and is always drawn into a certain side, the "positive" one. As regards the line, a certain direction will be called "positive", and namely we note, when we are dealing with the border line s of a surface $\sigma$, the following rule: If P is a fixed point of $\sigma$, very near to s, and if a second point Q traverses the nearest part of s in positive direction, then the rotation of PQ shall correspond to the direction of the perpendicular to $\sigma$.

As regards a closed surface, the outer side shall be positive.

d. Usually we denote vectors by German letters; these sometimes also serve to denote the magnitude only. By ${\mathfrak {A}}_{l}$ we understand the component of the vector ${\mathfrak {A}}$ into the direction l; by ${\mathfrak {A}}_{x},{\mathfrak {A}}_{y},{\mathfrak {A}}_{z}$ therefore the components into the axis-directions.