This paper discusses the problem of finding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of a sphere. A random search method is developed which uses a modification of the linearized equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable, though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.

中文翻译：

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球面上四点涡的平衡与稳定性

本文讨论了在球体表面寻找四点涡流的平衡位置的问题，这些涡流通常是不等循环的。开发了一种随机搜索方法，该方法使用线性化方程的修改来收敛于不同的平衡点。对于规定的循环和角冲量，可能存在许多平衡（47 和可能更多）。线性稳定性分析表明它们通常是不稳定的，但确实存在稳定的平衡。总的来说，平衡的多样性令人惊讶，包括那些围绕与角脉冲矢量相反的轴旋转的平衡。