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. In order to find the geometrical meanings of results in the theory obtained in use of the extensor analysis, one should make clear the relation between the geome tries in the space of higher order and in the original point space. By this reason, it seems to be desirable that the geometrical theory stated in words of the extensor analysis in the space of higher order is an extension of that of tensor analysis in the original point space and the part corresponding to tensor member of the former just coincides with the latter, so as an extensor introduced by H. V. Craig [1] 1) is an ex tension of a tensor and the components of the extensor include the tensor member as a part. As the first step, in the previous paper [2], the present author obtained already the extended connection parameters in the above-mentioned sense, in the case that either the original point space is a space with a symmetric affine connec tion or a Riemannian space. It is the purpose of the present paper to establish the theory of non-holonomic systems and subspaces in the space of higher order based upon a point space with a non-holonomic system ƒÉi'i, at the same standpoint. In the present paper, the ideas, notations and results given in the previous papers ([2] [5]) will be used without explanation. The present author wishes to ex press to Prof. A. Kawaguchi her appreciation for his helpful guidance.

Specialization of the Theory of a Space of Higher Order I