Let particle P be described by a chain of events

$\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \ \ldots \ \sf{P}_{\it{k}} \ \ldots \ \right)$

where each event is characterized by its displacement vector

$d\bar{r} = \left( dx, dy, dz \right)$

Definition: the **abscissa** of event $k$ is

$\begin{align} x_{k} \equiv x_{0} + \sum_{i=1}^{k} dx_{i} \end{align}$

where $x_{0}$ is arbitrary and often set to zero. The **ordinate** is

$\begin{align} y_{k} \equiv y_{0} + \sum_{i=1}^{k} dy _{i} \end{align}$

And the $z$-cooordinate or **applicate** of event $k$ is defined as

$\begin{align} z_{k} \equiv z_{0} + \sum_{i=1}^{k} dz_{i} \end{align}$

The $z$-component of the displacement $dz$ is a simple linear function of the wavelength. So if the particle is isolated then P moves in regular steps along the polar-axis as

$z_{k} = z_{0} + k dz$

The three numbers $x$, $y$ and $z$ are called the**Cartesian coordinates**of event $k$ after the work of René Descartes. More exactly, they are the

*rectangular*Cartesian coordinates in a descriptive system that is

*centered*on P. We use them to express the position of an event as

$\overline{r} = ( x, y, z )$

where $\overline{r}_{0} = ( x_{0}, y_{0}, z_{0} )$.