A quantum-mechanical non-relativistic spinnig particle is represented equivalently with a non-linear vector field which defines a new kind of hydrodynamics of a spinning fluid. In this hydrodynamics, quantum effects are separated as non-linear terms which mean the occurrence of “internal potential” and “internal magnetic field”. Mathematically, the method is based upon the replacement of the original calculus in terms of a spinor ψ by the calculus in terms of tensors formed as the bilinear expressions in ψ*. This article (hereafter called I) and also the major part of the subsequent one (hereafter called II) which will appear in succession, are the translation of the paper which appeared in Soryusiron-Kenkyu 7, pp. 600–640, (January 1955) in Japanese, while a brief account of these works was reported in Prog. Theor. Phys. 12 (1954), 810. Very recently articles by Bohm, Schiller, and Tiomno appeared (Nuovo Cimento 1, Suppl. (1955), 48, 67), in which they investigated, independently of us, the similar hydrodynamical method for the purpose of a “ causal interpretation ” of quantum theory in the case of a non-relativistic spinning particle. However, their method stands upon Euler angle representation of the spinor (cf II), not directly concerned with the self-contained tensor formulation of the spinor equation and ψ. This replacement is ensured by setting up the identities which should hold among those tensor quantities.