I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience. In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen. The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region. X₁, X₂, X₃ are the spatial co-ordinates; X₄ is the corresponding time-co-ordinate measured by some suitable measuring clock. These co-ordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure). The expression (1) _ds²_ = - _d_X₁² - _d_X₂² - _d_X₃² + _d_X₄² had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system. Let us take _ds_ as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If _ds²_ belonging to the element (_d_X₁, _d_X₂, _d_X₃, _d_X₄) be positive we call it with Minkowski, time-like, and in the contrary case space-like. To the line-element considered, _i.e._, to both the infinitely near point-events belong also definite differentials _dx₁_, _dx₂_, _dx₃_, _dx₄_, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a local system of the above kind given for the case under consideration, _d_X’s would then be represented by definite linear homogeneous expressions of the form (2) _d_X_{ν} = σ_{σ}_a__{νσ}_dx__{σ} If we substitute the expression in (1) we get (3) _ds²_ = σ_{στ}_g__{στ}_dx__{σ}_dx__{τ} where _g__{στ} will be functions of _x__{σ}, but will no longer depend upon the orientation and motion of the ‘local’ co-ordinates; for _ds²_ is a definite magnitude belonging to two point-events infinitely near in space and time and can be got by measurements with rods and clocks. The _g__{τσ}’s are here to be so chosen, that _g__{τσ} = _g__{στ}; the summation is to be extended over all values of σ and τ, so that the sum is to be extended, over 4 × 4 terms, of which 12 are equal in pairs. From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviour of _g__{στ} in a _finite_ region it is possible to choose the system of co-ordinates in such a way that _g__{στ} assumes constant values—