The concept of the non-holonomic system in the higher order space has been already given by the present author $[i]^{(1)}$ , and many operations in the system has been studied too [2] . It is Purpose of the present paper to treat the theory of. extensors in a subspace of the higher.order space under such the concept of the non-holonomic system. That is, we study, in \S 2 the operations introduced by A. KAWAGUCm [3] in the exsurface and the expseudonormal defined in \S 1 and give the $D$

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... of these operations. The same discussion is made for the excovariant differentiation in the sPace of the connection in \S \S 3-4. In this paper we use certain of the ideas, notations and results given in the previous paper [1] without explanation.