The ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity. While now I want to introduce geometrical figures in the manifold of the variables (_x_, _y_, _z_, _t_), it may be convenient to leave (_y_, _z_) out of account, and to treat _x_ and _t_ as any possible pair of co-ordinates in a plane, referred to oblique axes. A space time null point 0 (_x_, _y_, _z_, _t_ = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation. The figure -_x²_ - _y²_ - _z²_ + _t²_ = 1, _t_ > 0 ... (2) which represents a hyper boloidal shell, contains the space-time points A (_x_, _y_, _z_, _t_ = 0, 0, 0, 1), and all points A′ which after a Lorentz-transformation enter into the newly introduced system of reference as (_x′_, _y′_, _z′_, _t′_ = 0, 0, 0, 1). The direction of a radius vector 0A′ drawn from 0 to the point A′ of (2), and the directions of the tangents to (2) at A′ are to be called normal to each other. Let us now follow a definite position of matter in its course through all time _t_. The totality of the space-time points (_x_, _y_, _z_, _t_) which correspond to the positions at different times _t_, shall be called a space-time line. The task of determining the motion of matter is comprised in the following problem:—It is required to establish for every space-time point the direction of the space-time line passing through it. To transform a space-time point P (_x_, _y_, _z_, _t_) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference (_x′_, _y′_, _z′_, _t′_), in which the _t′_ axis has the direction 0A′, 0A′ indicating the direction of the space-time line passing through P. The space _t′_ = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P. To the increment _dt_ of the time of P corresponds the increment _d_τ = √(_dt²_ - _dx²_ - _dy²_) - _dz²_ = _dt_√(1 - _u²_) of the newly introduced time parameter _t′_. The value of the integral ∫ _dτ_ = ∫ √(-(_dx₁²_ + _dx₂²_ + _dx₃²_ + _dx₄²_)) when calculated upon the space-time line from a fixed initial point P₀ to the variable point P, (both being on the space-time line), is known as the ‘Proper-time’ of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.) If we take a body R₀ which has got extension in space at time _t₀_, then the region comprising all the space-time line passing through R₀ and _t₀_ shall be called a space-time filament. If we have an analytical expression θ(_x_ _y_, _z_, _t_) so that θ(_x_, _y_ _z_ _t_) = 0 is intersected by every space time line of the filament at one point,—whereby -(∂Θ/∂_x_)², -(∂Θ/∂_y_)², -(∂Θ/∂_z_)², -(∂Θ/∂_t_)² > 0, ∂Θ/∂_t_ > 0.