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Geometry in the manifold Kn(m) of line elements of a higher order, that is, (at, dx, ■ ■ ■ ,dmx) or {x,x', ■ ■ ■ , at(m)), in a space of «dimensions with a point coordinate system at*', {i = 1, 2, • ■ -, «), was first furnished with its foundation by the present author.f In close connection with this theory, there are two important problems. The first is the geometry of paths in this space, that is, the geometrical study of the system of paths xi = xi{t) which are defined by the ordinary differential equations of the form

Geometry in an 𝑛-dimensional space with the arc length 𝑠=∫{𝐴ᵢ(𝑥,𝑥’)𝑥”ⁱ+𝐵(𝑥,𝑥’)}^{1/𝑝}𝑑𝑡