that must vary in an appropriate way as we consider various portions of the sphere’s surface. In fact, owing to the constancy of , we must have proportional to . If, then, the meridians were parallel lines instead of tapering together towards the poles, would remain constant. Whence it follows that a knowledge of the way varies from point to point yields us information on the shape of the mesh-system, and inversely the lay of the mesh-system yields us information as to the value of from place to place. Then again, if, leaving our two points and fixed, we select some new mesh-system, may change in value, and in fact a difference may also appear, since with a change of mesh-system there is no reason why and should still lie on the same curve. Hence we must conclude that even at a fixed point of our surface the value of will be subject to change when we vary our mesh-system. Precisely the same arguments would apply were we to consider the expression of the distance between two infinitely close points on the same curve. We should then obtain , where is a magnitude generally differing from . Fig. III We must now consider the more general case where our two points lie on different and curves (Fig. III). From what precedes we know that the squared distance is given by and that the squared distance is expressed by . But this information is obviously insufficient to tell us what the squared distance will be, since this distance will also be affected by the slant of the lines and . Some new magnitude will obviously have to be introduced in order to specify the value of this slant; and the new magnitude considered will be called and . Under the circumstances, the squared distance of the infinitesimally near points and can be written but as and are found to be always identical we have We may write this formula more concisely by referring to as and to as . In this case our formula becomes where indicates summation and where for and we substitute the values 1 and 2 in all possible ways. This most important mathematical expression was discovered by Gauss; it is destined [Pg 88] to play a part of paramount significance in the physical theory of relativity. In order to understand the geometrical meaning of , let us consider the special case of a diamond-shaped mesh-system, where the co-ordinate lines make an angle (Fig. IV). Elementary geometry teaches us that Fig. IV The lines being always equally spaced, and remain constant throughout the mesh-system; hence we may put them equal to unity and write In this formula we have from which we see that represents the cosine of the angle formed by the and lines, at the point where is calculated.[26] Inasmuch as , we see that where the and lines are perpendicular, vanishes. Hence, whenever we have an expression of , such as