The transformation of temperature for a moving observer does not often \index{Temperature}% concern us. In general the word obviously means proper-temperature, and the motion of the observer does not enter into consideration. In thermometry and in the theory of gases it is essential to take a standard with respect to which the matter is at rest on the average, since the indication of a thermometer moving rapidly through a fluid is of no practical interest. But thermodynamical temperature is defined by \[ dS = dM/T, \Tag{(14.3)} \] where $dS$~is the change of entropy for a change of energy~$dM$. The temperature~$T$ \index{Entropy}% defined by this equation will depend on the observer's frame of reference. Entropy is clearly meant to be an invariant, since it depends on the probability of the statistical state of the system compared with other states which might exist. Hence $T$~must be altered by motion in the same way as~$dM$, that is to say \[ T' = \beta T. \Tag{(14.4)} \] But it would be useless to apply such a transformation to the adiabatic gas-equation \[ T = k\rho^{\gamma-1}, \] for, in that case, $T$~is evidently intended to signify the proper-temperature and $\rho$~the proper-density. In general it is unprofitable to apply the Lorentz transformation to the \emph{constitutive equations} of a material medium and to coefficients occurring in \index{Constitutive equations}% them (permeability, specific inductive capacity, elasticity, velocity of sound). Such equations naturally take a simpler and more significant form for axes moving with the matter. The transformation to moving axes introduces great complications without any evident advantages, and is of little interest except as an analytical exercise. \Section{15.}{General transformations of coordinates} \index{Coordinates!general transformation of}% We obtain a transformation of coordinates by taking new coordinates \index{Transformation of coordinates!general}% $x_{1}'$, $x_{2}'$, $x_{3}'$, $x_{4}'$ which are any four functions of the old coordinates $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. Conversely, $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ are functions of $x_{1}'$, $x_{2}'$, $x_{3}'$,~$x_{4}'$. It is assumed that \PageSep{35} multiple values are excluded, at least in the region considered, so that values of $(x_{1}, x_{2}, x_{3}, x_{4})$ and $(x_{1}', x_{2}', x_{3}', x_{4}')$ correspond one to one.