It requires a greater impulse to produce a given change of velocity~$\delta v$ in the original direction of motion than to produce an equal change~$\delta w$ at right angles to it. For the momenta in the two directions are initially \[ mv(1 - v^{2}/c^{2})^{-\frac{1}{2}},\quad 0, \] and after a change $\delta v$, $\delta w$, they become %[** TN: Not broken in the original] \begin{gather*} m(v + \delta v) \bigl[1 - \bigl\{(v + \delta v)^{2} + (\delta w)^{2}\bigr\}/c^{2}\bigr]^{-\frac{1}{2}},\displaybreak[0] \\ m\, \delta w \bigl[1 - \bigl\{(v + \delta v)^{2} + (\delta w)^{2}\bigr\}/c^{2}\bigr]^{-\frac{1}{2}}. \end{gather*} Hence to the first order in $\delta v$, $\delta w$\Add{,} the changes of momentum are \[ m(1 - v^{2}/c^{2})^{-\frac{3}{2}}\, \delta v,\quad m(1 - v^{2}/c^{2})^{-\frac{1}{2}}\, \delta w, \] or \[ M\beta^{2}\, \delta v,\quad M\, \delta w, \] where $\beta$~is the FitzGerald factor for velocity~$v$. The coefficient $M\beta^{2}$ was formerly called the \emph{longitudinal mass}, $M$~being the \emph{transverse mass}; but the \index{Longitudinal mass}% longitudinal mass is of no particular importance in the general theory, and the term is dropping out of use. \PageSep{32} \Section{13.}{Energy} When the units are such that $c = 1$, we have \begin{align*} M &= m(1 - v^{2})^{-\frac{1}{2}} \\ &= m + \tfrac{1}{2} mv^{2} \text{ approximately,} \Tag{(13.1)} \end{align*} if the speed is small compared with the velocity of light. The second term is the kinetic energy, so that the change of mass is the same as the change of \index{Energy, identified with mass}% \index{Mass!identified with energy}% energy, when the velocity alters. This suggests the identification of mass with energy. It may be recalled that in mechanics the total energy of a system is left vague to the extent of an arbitrary additive constant, since only changes of energy are defined. In identifying energy with mass we fix the additive constant~$m$ for each body, and $m$~may be regarded as the internal energy of constitution of the body. The approximation used in~\Eq{(13.1)} does not invalidate the argument. Consider two ideal billiard balls colliding. The conservation of mass (relative \index{Conservation!of energy}% mass) states that \[ \sum m (1 - v^{2})^{-\frac{1}{2}} \text{ is unaltered.} \] The conservation of energy states that \[ \sum m (1 + \tfrac{1}{2}v^{2}) \text{ is unaltered.} \] But if both statements were exactly true we should have two equations determining unique values of the speeds of the two balls; so that these speeds could not be altered by the collision. The two laws are not independent, but one is an approximation to the other. The first is the accurate law since it is independent of the space-time frame of reference. Accordingly the expression $\frac{1}{2}mv^{2}$ for the kinetic energy in elementary mechanics is only an approximation in which terms in~$v^{4}$, etc.\ are neglected.