For the discussion of space and time we have made use of certain ideal apparatus which can only be imperfectly realised in practice---rigid scales and \PageSep{30} perfect cyclic mechanisms or clocks, which always remain similar configurations from the absolute point of view. Similarly for the discussion of inertia we require some ideal material object, say a perfectly elastic billiard ball, whose condition as regards inertial properties remains constant from an absolute point of view. The difficulty that actual billiard balls are not perfectly elastic must be surmounted in the same way as the difficulty that actual scales are not rigid. To the ideal billiard ball we can affix a constant number, called the \emph{invariant mass}\footnotemark,\footnotetext {Or \emph{proper-mass}.} which will denote its absolute inertial properties; and \index{Invariant}% \index{Invariant mass}% \index{Mass!invariant and relative}% \index{Mass!variation with velocity}% this number is supposed to remain unaltered throughout the vicissitudes of its history, or, if temporarily disturbed during a collision, is restored at the times when we have to examine the state of the body. With the customary definition of momentum, the components \[ M\, \frac{dx}{dt},\quad M\, \frac{dy}{dt},\quad M\, \frac{dz}{dt} \Tag{(12.1)} \] cannot satisfy a general law of conservation of momentum unless the mass~$M$ \index{Conservation!of momentum and mass}% is allowed to vary with the velocity. But with the slightly modified definition \[ m\, \frac{dx}{ds},\quad m\, \frac{dy}{ds},\quad m\, \frac{dz}{ds} \Tag{(12.2)} \] the law of conservation can be satisfied simultaneously in all space-time systems, $m$~being an invariant number. This was shown in \Title{Space, Time and Gravitation}, p.~142. Comparing \Eq{(12.1)} and \Eq{(12.2)}, we have \[ M = m\, \frac{dt}{ds}. \Tag{(12.3)} \] We call $m$ the \emph{invariant mass}, and $M$ the \emph{relative mass}, or simply the \emph{mass}. The term ``invariant'' signifies unchanged for any transformation of coordinates, and, in particular, the same for all observers; constancy during the life-history of the body is an additional property of~$m$ attributed to our ideal billiard balls, but not assumed to be true for matter in general. Choosing units of length and time so that the velocity of light is unity, we have by~\Eq{(7.2)} \[ \frac{ds}{dt} = (1 - v^{2})^{\frac{1}{2}}. \] Hence by \Eq{(12.3)} \[ M = m(1 - v^{2})^{-\frac{1}{2}}. \Tag{(12.4)} \] The mass increases with the velocity by the same factor as that which gives the FitzGerald contraction; and when $v = 0$, $M = m$. The invariant mass is thus equal to the mass at rest.