Proceeding by the same method as we did in the case of two dimensions, we find that there now exist several invariant types of relations between the ’s at every definite point and their variations in value at all points near this one. The values of these expressions remain unchanged when we alter our mesh-system; hence, as in the case of the Gaussian curvature for two-dimensional space, they refer to the geometry of the space itself and not to our choice of mesh-system. One of these invariant expressions is the generalisation of the Gaussian curvature extended to four dimensions. The others refer to new types of curvatures which appear only in spaces of more than two dimensions. It is impossible to represent these curvatures with a two-dimensional surface because they then become identified with the ordinary Gaussian curvature.[31] For this reason it was only when curved spaces of more than two dimensions were studied that these new forms of curvature were brought into prominence. [Pg 97] From what has been said we may anticipate marked complications when we wish to study the geometry of a space of more than two dimensions. Thus, we saw that in a two-dimensional space the Gaussian curvature fully defines the geometry of the space. For instance, if the Gaussian curvature vanishes throughout, the surface is Euclidean or at least flat.[32] In the same way, if the Gaussian curvature is an invariable positive or negative number throughout, the surface is one of constant curvature, either positive, as with a sphere, or negative, as with a pseudosphere. But when we consider spaces of more than two dimensions, a knowledge of the generalised Gaussian curvature throughout the space is no longer sufficient to fix its geometry. While this curvature may vanish or present the same non-vanishing value throughout, we cannot infer therefrom that the space is necessarily flat or of constant curvature. Thus, whereas the vanishing of the generalised Gaussian curvature is a necessary condition for the space to be flat, it is by no means sufficient. There is still room for a large measure of indeterminateness in the actual geometry of the space.