  Higher-order photon rings can be expected to be detected in a more detailed
image of the black hole found in future observations. These rings are lensed
images of the luminous matter surrounding the black hole and are formed by
photons that loop around it. In this paper we have succeeded to derive an
analytical expression for the shape of the higher-order rings in the form that
is most convenient for application: the explicit equation of the curve in polar
coordinates. The formula describes the apparent shape of the higher-order image
of the circular ring with the given radius around Schwarzschild black hole as
viewed by distant observer with an arbitrary inclination. For the derivation,
the strong deflection limit of the gravitational deflection is used. Our
formula is a simple and efficient alternative to the numerical calculation of
ray trajectories, with the main application to studying the shape of $n=2$ and
$n=3$ photon rings.
