This paper aims, in the theory of Dirac electron, at the deliberate use of the hyperbolic metric of velocity, the effectiveness of which Smorodinski1 has recently emphasized in relativity. The metric of velocity is fully used together w~th a proper spin direction vector, to represent explicitly spinor cf; and vector, tensor, etc., of cf;* (matrix) cf;, of free Dirac electron. The 3-vector polarization operator is confirmed to correspond to the proper spin vector. For the electron in a square well potential, cf; is a product of hyperbolic function of the metric with trigonometric or the Bessel function, etc., and the metric turns out to be imaginary or complex in the case of the bound state or free negative energy state. A classical model of Dirac electron is established concretely by obtaining simple time derivatives of the metric and spin vector, from the covariant equations of a spinning particle under general fields, suggested by Frenkel, Kramers and others. The so-called Thomas factor 1/2 is brought inevitably in the equation of the spin vector itself, regardless of the time average motion. These explicit representations may facilitate the visualization of Dirac electron. §I. IntroductionThe hyperbolic metric V of velocity, the so-called rapidity, may be expressed asfor the relativistic velocity v = dr/ dt. In the representation of the Lorentz transformation, the imaginary rotation angle or hyperbolic one which has been preferably used in the forms of cosh V, sinh V and cosh (V /2), sinh (V /2), etc., stands exactly for this metric. The significance of the usage of this metric has been clarified in the old text of Pauli 1 ) on relativity, and also in the recent one of F ock. 2 ) In 1944, Karapetoff 3 l used actually this metric in the regions of relativistic mechanics and electrodynamics. More recently, Srnorodinski1 4 l has explained that the relativistic velocity space is isomorphic to the Lobachevskian space, namely, hyperbolic space, and has applied this fact to the calculations of collisions of particles. The conscious use of this metric, however, is not yet prevalent in most of many works on relativity, although the convenience of this *> Some parts of this paper are the revised translation of the author's work published in Rep.