from which is absent, we may be certain that the and lines are orthogonal at the point for which has been calculated. If the and lines remain perpendicular throughout the entire mesh-system, will vanish not only for one particular point, but for all points of the surface. Let us now consider the particular case of a Cartesian mesh-system of equal squares. In this case, the and lines being orthogonal [Pg 89] and equally spaced, we have and constant, with vanishing. Hence we may replace and by unity, and we obtain a result in complete agreement with the Pythagorean theorem of the square of the hypothenuse. Fig. V Suppose, now, that on this same plane we were to trace another type of co-ordinate system, one known as a polar system (Fig. V). It is constituted by symmetrically disposed v lines radiating from a central point . The lines are then given by equally distanced concentric circles having as centre. Under the circumstances the expression for becomes In other words, we have proving once again that the values of the ’s vary with our choice of a co-ordinate system. As before, vanishes for the same reasons as previously stated, namely, because the lines of our mesh-system are orthogonal at their intersections. Finally, we may consider the mesh-system defined by the parallels and meridians on a sphere, where we may assume that gives a parallel and gives a meridian. Here we have so that From these various examples the following conclusions may be drawn: 1°. When a definite mesh-system has been traced on a surface and its lines numbered consecutively, every point of the surface is defined unambiguously by its two numbers (its two Gaussian numbers or co-ordinates). These numbers represent, of course, those of the [Pg 90] respective and curves at whose intersection the point stands. A variation in the shape or in the numbering of the and lines entails a change in the co-ordinate numbers of any given point on the surface. 2°. Alongside of these two Gaussian numbers at every point defining the positions of points in the mesh-system, there exist four numbers, at every point, namely , , , . But as and are always identical, we have only three of these magnitudes to consider. The values of these ’s at a point vary when we change our mesh-system. In the majority of cases they also vary from place to place throughout the same mesh-system. If, however, the meshes are always orthogonal at their points of intersection, vanishes at every point of the surface. If the and lines always have the same slant over the surface and are always equally spaced, the three ’s will remain constant throughout; their values being given by 1, , 1. If the mesh-system is Cartesian, hence forms equal squares , of course, vanishes while, as before, and remain constantly equal to unity. 3°. The value of the square of the distance between two infinitesimally distant points is given by