Page:Eddington A. Space Time and Gravitation. 1920.djvu/94

system are you using? No one can form a picture of the triangle until that information has been given. But if we speak of the properties of a triangle whose sides are of lengths 2, 3, 4 inches, anyone with a graduated scale can draw the triangle, and follow our discussion of its properties. The distance between two points can be stated without referring to any mesh-system. For this reason, if we use a mesh-system, it is important to find formulae connecting the absolute distance with the particular system that is being used.

In the more complicated kinds of mesh-systems it makes a great simplification if we content ourselves with the formulae for very short distances. The mathematician then finds no difficulty in extending the results to long distances by the process called integration. We write $ds$ for the distance between two points

close together, $x_{1}$ and $x_{2}$ for the two numbers specifying the location of one of them, $dx_{1}$ and $dx_{2}$ for the small differences of these numbers in passing from the first point to the second. But in using one of the particular mesh-systems illustrated in the diagrams, we usually replace $x_{1}$, $x_{2}$ by particular symbols sanctioned by custom, viz. $(x_{1},x_{2})$ becomes $(x,y)$, $(r,\theta )$, $(\xi ,\eta )$ for Figs. 10, 11, 12, respectively.

The formulae, found by geometry, are:

For rectangular coordinates $(x,y)$, Fig. 10, $ds^{2}=dx^{2}+dy^{2}$. For polar coordinates $(r,\theta )$, Fig. 11, $ds^{2}=dr^{2}+r^{2}d\theta ^{2}$. For oblique coordinates $(\xi ,\eta )$, Fig. 12, $ds^{2}=d\xi ^{2}-2\kappa \,d\xi \,d\eta +d\eta ^{2}$, where $\kappa$ is the cosine of the angle between the lines of partition.