In a stratified or a spherically symmetric atmosphere, we consider some cases where a curved light ray connects two points, $A$A and $S$S, on the sea or the ground, at the same height as in some cases of superior mirages or "arctic mirages." We study the range $D$D of this ray (i.e., the geodesic distance between $A$A and $S$S) as a function of ${\alpha _S}$αS, the ray inclination with respect to the vertical at $A$A or $S$S. The fundamental expression of $D$D is an improper integral. We show that a change of variable, also used to calculate the astronomical refraction, usually gives a well-behaved integral. In a spherically symmetric atmosphere, this happens when the refraction coefficient exceeds unity on the entire ray, producing a hillingar effect (alias strong looming for an erect image) or superior mirages. We then find a nice expression of ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dαS, which is usually negative. Inverted images (i.e., mirages in the strict sense) show up when ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dαS is positive; for this, we find a necessary condition in each atmospheric model. The strength of these results comes from not requiring precise knowledge of the ray paths.

中文翻译：

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关于“北极海市ages楼”的一些定理：希林加效应和高级海市ages楼。

在分层或球形对称的大气中，我们考虑了某些情况，其中弯曲的光线在海或地面上以与某些上等海市ages楼相同的高度连接两个点，即$ A $ A和$ S $ S或“北极海市ages楼”。我们研究了该射线的范围$ D $ D（即$ A $ A和$ S $ S之间的测地距离）与$ {\ alpha _S} $αS的关系，即射线相对于垂直方向的倾斜度$ A $ A或$ S $ S。$ D $ D的基本表达形式是不正确的积分。我们表明，变量的变化（也用于计算天文折射）通常会提供良好的积分。在球对称大气中，当折射系数在整个光线上超过1时会发生这种情况，从而产生希林加效应（对于正像，通常会出现强烈的隐约可见）或上等的幻影。然后，我们发现一个很好的表达式$ {{{\ rm d} D} / {{\ rm d} {\ alpha _S}}} $ dD /dαS，通常为负数。当$ {{{\ rm d} D} / {{\ rm d} {\ alpha _S}}} $ dD /dαS为正值时，出现反转图像（即严格意义上的幻影）。为此，我们在每个大气模型中找到了必要条件。这些结果的强度来自于不需要精确了解射线路径的原因。