$$ y = \frac {(1-\frac {v^2}{c^2})^{\frac {1}{2}} w_{\eta}t} {1+\frac {vw_{\xi}}{c^2}} $$ , z = 0 . The law of parallelogram of velocities hold up to the first order of approximation. We can put $$ U^2 = (\frac {\partial x}{\partial t})^2 + (\frac {\partial y}{\partial t})^2 $$ , $$ w^2 = w_{\xi}^2 + w_{\eta}^2 $$ , and $$ \alpha = tan^{-1} \frac {w}{w_{\xi}} $$ _i.e._, α is put equal to the angle between the velocities _v_, and _w_. Then we have— $$ U = \frac {[(v^2 + w^2 + 2 vw \cos \alpha) - (\frac {vw \sin \alpha}{c})^2]^{\frac {1}{2}}} {1 + \frac {vw \cos \alpha}{c^2}} $$ It should be noticed that _v_ and _w_ enter into the expression for velocity symmetrically. If _w_ has the direction of the ξ-axis of the moving system, $$ U = \frac {v + w}{1 + \frac {vw}{c^2}} $$ From this equation, we see that by combining two velocities, each of which is smaller than _c_, we obtain a velocity which is always smaller than _c_. If we put _v_ = _c_ - χ, and _w_ = _c_ - λ, where χ and λ are each smaller than _c_,[8] $$ U = c \frac {2c - \chi - \lambda}{2c - \chi - \lambda + \frac {\chi \lambda}{c^2}} < c $$ It is also clear that the velocity of light _c_ cannot be altered by adding to it a velocity smaller than _c_. For this case, $$ U = \frac {c + v}{1 + \frac {cv}{c^2}} = c $$ We have obtained the formula for U for the case when _v_ and _w_ have the same direction; it can also be obtained by combining two transformations according to section § 3. If in addition to the systems K, and k, we introduce the system k´, of which the initial point moves parallel to the ξ-axis with velocity _w_, then between the magnitudes, _x_, _y_, _z_, _t_ and the corresponding magnitudes of k´, we obtain a system of equations, which differ from the equations in § 3, only in the respect that in place of _v_, we shall have to write, $$ \frac {v + w}{1 + \frac {vw}{c^2}} $$ We see that such a parallel transformation forms a group. We have deduced the kinematics corresponding to our two fundamental principles for the laws necessary for us, and we shall now pass over to their application in electrodynamics. II.—ELECTRODYNAMICAL PART. § 6. Transformation of Maxwell’s equations for Pure Vacuum. On the nature of the Electromotive Force caused by motion in a magnetic field. The Maxwell-Hertz equations for pure vacuum may hold for the stationary system K, so that $$ \frac {1}{c} \frac {\partial}{\partial t} [X, Y, Z] = \begin{vmatrix} \frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac {\partial}{\partial z} L & M & N \end{vmatrix} $$ and $$ \frac {1}{c} \frac {\partial}{\partial t} [L, M, N] = \begin{vmatrix} \frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac {\partial}{\partial z} X & Y & Z \end{vmatrix} $$ (1) where [X, Y, Z] are the components of the electric force, L, M, N are the components of the magnetic force.