XXI.—Relative Co-ordinates

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Abstract
If C is a given curve in a Riemannian space Vn, a system of co-ordinates (z1, z2, …, zn) can be set up at each point P of C, thus generalising moving axes along a twisted curve. If in these co-ordinates a point Q is defined relative to the point P, then Q traces some curve as P moves along C; any curve C' can be defined in this way by setting up a (1, 1) correspondence between points of C and C′. We shall take the relative co-ordinates at P to be normal co-ordinates with origin at P, the parametric directions at P being the directions of an orthogonal ennuple defined at points of C. In general, this work is too heavy except for the consideration of points within a certain distance from the curve C, and we shall therefore consider only points the cube of whose distance from C may be neglected. This is sufficient for the application to such problems as the motion of a small rigid body in space-time.

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