= Adjacent-order spacing versus nearest-axis echelle interval
{title2=$S_{\rm centres}=K/[m(m+1)]$}
At a fixed camera-axis direction, the <grating equation> gives $m\lambda_m=K=d(\sin\alpha+\sin\beta_0)$, so adjacent central <wavelengths> differ by $K/[m(m+1)]$. This differs from the exact <wavelength> cell where an order is nearest the <optical camera> axis. For $\beta_0=0$ and $x=f\tan\beta$, equal distances of orders $m$ and $m+1$ mean opposite <diffraction> <angles>; adding their <diffraction grating> equations gives $\lambda=K/(m+1/2)$. The full order-$m$ cell is bounded by $K/(m+1/2)$ and $K/(m-1/2)$, hence has width $K/(m^2-1/4)$ for $m\ge2$ and accessible neighbouring orders. Both widths approach $K/m^2$ at high order.
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