XXV. Gaussian Co-ordinates

Gaussian Co-ordinates

According to Gauss, this combined analytical

and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see [fig:4]Fig. 4) drawn on the surface of the table. These we designate as $u$ -curves, and we indicate each of them by means of a number. The curves $u = 1$ , $u = 2$ and $u = 3$ are drawn in the diagram. Between the curves $u = 1$ and $u = 2$ we must imagine an infinitely large number to be drawn, all of which correspond

2in087 to real numbers lying between $1$ and $2$ . We have then a system of $u$ -curves, and this "infinitely dense" system covers the whole surface of the table. These $u$ -curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly definite value of $u$ belongs to every point on the surface of the marble slab. In like manner we imagine a system of $v$ -curves drawn on the surface. These satisfy the same conditions as the $u$ -curves, they are provided with numbers

in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of $u$ and a value of $v$ belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates).

For example, the point $P$ in the diagram has the Gaussian co-ordinates $u = 3$ , $v = 1$ . Two neighbouring points $P$ and $P^{'}$ on the surface then correspond to the co-ordinates

where $d u$ and $d v$ signify very small numbers. In a similar manner we may indicate the distance (line-interval)

between $P$ and $P^{'}$ , as measured with a little rod, by means of the very small number $d s$ . Then according to Gauss we have $d s^{2} = g_{11} d u^{2} + 2 g_{12} d u d v + g_{22} d v^{2},$ where $g_{11}$ , $g_{12}$ , $g_{22}$ , are magnitudes which depend in a perfectly definite way on $u$ and $v$ . The magnitudes $g_{11}$ , $g_{12}$ and $g_{22}$ determine the behaviour of the rods relative to the $u$ -curves and $v$ -curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum

with reference to the measuring-rods, but only in this case, it is possible to draw the $u$ -curves and $v$ -curves and to attach numbers to them, in such a manner, that we simply have: $d s^{2} = d u^{2} + d v^{2} .$ Under these conditions, the $u$ -curves and $v$ -curves are straight lines in the sense of Euclidean geometry, and

they are perpendicular to each other. Here the Gaussian co-ordinates are simply Cartesian ones. It is clear

that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points "in space."

So far, these considerations hold for a continuum