The general theory of curvature of curves and surfaces may in a similar way be regarded as a theory of their invariants for the group of motions. That something more than a mere change of phraseology is here implied will be evident in dealing with minimum curves, i.e. with curves such that at every point of them dx^2 + dy^2 + dz^2 = 0. For such curves the ordinary theory of curvature has no meaning, but they nevertheless have invariant properties in regard to the group of motions. The curvature and torsion of a curve, which are invariant for all transformations by the group of motions, are special instances of what are known as _differential invariants_. If [xi]([Pd]/[Pd]x) + [eta]([Pd]/[Pd]y) is the general infinitesimal transformation of a group of point-transformations in the plane, and if y1, y2, ... represent the successive differential coefficients of y, the infinitesimal transformation may be written in the extended form [Pd] [Pd] [Pd] [Pd] [xi] ----- + [eta] ----- + [eta]1 ------ + [eta]2 ------ + ... [Pd]x [Pd]y [Pd]y1 [Pd]y2 where [eta]1[delta]t, [eta]2[delta]t, ... are the increments of y1, y2, .... By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.