Abstract
where F is any point function and ds is the element of arc in the space considered.f In the plane such a falnily contains oC2 curves, in ordinary space, the number is O04, and in the general case of n dimensions the number is x2f-l, one passing through every point in every direction. The results obtained hold for the general case though for simplicity they will be written out in the following only for three dimensions. Any system of curves connected with an integral (1) we shall refer to as a naturat system or family. The name seems justified in view of the many important geometrical and physical illvestigations wllich lead to variation problems of type (1). The following six are perhaps ths most noteworthy. 1) ltrajectories in a conservative field of force. Let a unit particle move in a field due to the work function (negative potential) W(Z? ys z). The totality of motions, for all initial conditions, given by the equations
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