To some extent we can avoid this failure by taking the microscopic point of view. The billiard ball can be analysed into a very large number of constituents---electrons and protons---each of which is believed to preserve the same invariant mass for life. But the invariant mass of the billiard ball is not exactly equal to the sum of the invariant masses of its constituents\footnotemark.\footnotetext {This is because the invariant mass of each electron is its relative mass referred to axes moving with it; the invariant mass of the billiard ball is the relative mass referred to axes at rest in the billiard ball as a whole.} The permanence and permanent similarity of all electrons seems to be the modern equivalent of Lavoisier's ``conservation of matter.'' It is still uncertain whether it expresses a universal law of nature; and we are willing to contemplate the possibility that occasionally a positive and negative electron may coalesce and annul one another. In that case the mass~$M$ would pass into the electromagnetic waves generated by the catastrophe, whereas the invariant mass~$m$ would disappear altogether. Again if ever we are able to synthesise helium out of hydrogen, $0.8$~per cent, of the invariant mass will be annihilated, whilst the corresponding proportion of relative mass will be liberated as radiant energy. It will thus be seen that although in the special problems considered the quantity~$m$ is usually supposed to be permanent, its conservation belongs to an altogether different order of ideas from the universal conservation of~$M$. \Section{14.}{Density and temperature} \index{Density!Lorentz transformation of}% Consider a volume of space delimited in some invariant way, e.g.\ the content of a material box. The counting of a number of discrete particles continually within (i.e.\ moving with) the box is an absolute operation; let the absolute number be~$N$. The volume~$V$ of the box will depend on the space-reckoning, being decreased in the ratio~$\beta$ for an observer moving relatively to the box and particles, owing to the FitzGerald contraction of one of the dimensions of the box. Accordingly the particle-density $\sigma = N/V$ satisfies \[ \sigma' = \sigma\beta, \Tag{(14.1)} \] \PageSep{34} where $\sigma'$~is the particle-density for an observer in relative motion, and $\sigma$~the particle-density for an observer at rest relative to the particles. It follows that the mass-density~$\rho$ obeys the equation \[ \rho' = \rho\beta^{2}, \Tag{(14.2)} \] since the mass of each particle is increased for the moving observer in the ratio~$\beta$. Quantities referred to the space-time system of an observer moving with \index{Proper-(prefix)}% the body considered are often distinguished by the prefix \emph{proper-} (German, \emph{Eigen-}), e.g.\ proper-length, proper-volume, proper-density, proper-mass $=$~invariant mass.