It is of some interest to examine in detail how this difference of reckoning of simultaneity arises. It has been explained in \SecRef{4} that by convention the time at two places is compared by transporting a clock from one to the other \index{Clocks, transport of}% \index{Transport of clocks}% with infinitesimal velocity. Our formulae are based on this convention; and, of course, \Eq{(11.1)}~will only be true if the convention is adhered to. The fact that infinitesimal velocity relative to~$S'$ is not the same as infinitesimal velocity relative to~$S$, leaves room for the discrepancy of reckoning of simultaneity to creep in. Consider two points $A$ and~$B$ at rest relative to~$S'$, and distant~$x'$ apart. Take a clock at~$A$ and move it gently to~$B$ by giving it an \PageSep{28} \index{Electromagnetic action!signals}% infinitesimal velocity~$du'$ for a time~$x'/du'$. Owing to the motion, the clock will by~\Eq{(4.9)} be retarded in the ratio $(1 - du'^{2}/c^{2})^{-\frac{1}{2}}$; this continues for a time~$x'/du'$ and the total loss is thus \[ \bigl\{1 - (1 - du'^{2}/c^{2})^{\frac{1}{2}}\bigr\} x'/du', \] which tends to zero when $du'$~is infinitely small. $S'$~may accordingly accept the result of the comparison without applying any correction for the motion of the clock. Now consider $S$'s~view of this experiment. For him the clock had already a velocity~$u$, and accordingly the time indicated by the clock is only $(1 - u^{2}/c^{2})^{\frac{1}{2}}$ of the true time for~$S$. By differentiation, an additional velocity~$du$\footnote {Note that $du$ will not be equal to~$du'$.} causes a supplementary loss \[ (1 - u^{2}/c^{2})^{-\frac{1}{2}} u\, du/c^{2} \text{ clock seconds} \Tag{(11.2)} \] per true second. Owing to the FitzGerald contraction of the length~$AB$, the distance to be travelled is~$x'/\beta$, and the journey will occupy a time \[ x'/\beta\, du \text{ true seconds}. \Tag{(11.3)} \] Multiplying \Eq{(11.2)} and \Eq{(11.3)}, the total loss due to the journey is \[ ux'/c^{2} \text{ clock seconds,} \] or \[ \beta ux'/c^{2} \text{ true seconds for~$S$.} \Tag{(11.4)} \] Thus, whilst $S'$~accepts the uncorrected result of the comparison, $S$~has to apply a correction~$\beta ux'/c^{2}$ for the disturbance of the chronometer through transport. This is precisely the difference of their reckonings of simultaneity given by~\Eq{(11.1)}.