Let us introduce for this purpose a third co-ordinate system _k′_, which is set in motion relative to the system _k_, the motion being parallel to the ξ-axis. Let the velocity of the origin be (-_v_). At the time _t_ = 0, all the initial co-ordinate points coincide, and for _t_ = _x_ = _y_ = _z_ = 0, the time _t′_ of the system _k′_ = 0. We shall say that (_x′_ _y′_ _z′_ _t′_) are the co-ordinates measured in the system _k′_, then by a two-fold application of the transformation-equations, we obtain _v_ τ′ = φ(-_v_)β(-_v_){τ + ----- ξ} _c²_ = φ(_v_)φ(-_v_)t, _x′_ = φ](_v_)β(_v_)(ξ + _v_τ) = φ(_v_)φ(-_v_)_x_, etc. Since the relations between (_x′_, _y′_, _z′_, _t′_), and (_x_, _y_, _z_, _t_) do not contain time explicitly, therefore K and _k′_ are relatively at rest. It appears that the systems K and _k′_ are identical. ∴ φ(_v_)φ(-_v_) = 1. Let us now turn our attention to the part of the ξ-axis between (ξ = 0, η = 0, ζ = 0), and (ξ = 0, η = 1, ζ = 0). Let this piece of the _y_-axis be covered with a rod moving with the velocity _v_ relative to the system K and perpendicular to its axis;—the ends of the rod having therefore the co-ordinates _x₁_ = _vt_, _y₁_ = _l_ / φ(_v_), _z₁_ = 0 _x₂_ = _vt_, _y₂_ = 0, _z₂_ = 0 Therefore the length of the rod measured in the system K is _l_/φ(_v_). For the system moving with velocity (-_v_), we have on grounds of symmetry, _l_ _l_ -------- = --------- φ(_v_) φ(-_v_) ∴ φ(_v_) = φ(-_v_), ∴ φ(_v_) = 1. § 4. The physical significance of the equations obtained concerning moving rigid bodies and moving clocks. Let us consider a rigid sphere (_i.e._, one having a spherical figure when tested in the stationary system) of radius R which is at rest relative to the system (K), and whose centre coincides with the origin of K then the equation of the surface of this sphere, which is moving with a velocity _v_ relative to K, is ξ² + η² + ζ² = R². At time _t_ = 0, the equation is expressed by means of (_x_, _y_, _z_, _t_,) as $$ \frac {x^2}{(\sqrt {1 - \frac {v_2}{c_2}})^2} + y^2 + z^2 = R^2. $$ A rigid body which has the figure of a sphere when measured in the moving system, has therefore in the moving condition—when considered from the stationary system, the figure of a rotational ellipsoid with semi-axes $$ R \sqrt {1 - \frac {v^2}{c^2}}, R, R. $$ Therefore the _y_ and _z_ dimensions of the sphere (therefore of any figure also) do not appear to be modified by the motion, but the _x_ dimension is shortened in the ratio $$ 1 : \sqrt {1 - \frac {v^2}{c^2}}; $$