We present a new class of electrostatics problems that are exactly solvable by adding finitely many image charges. Given a charge at some location inside a cavity bounded by up to four conducting grounded segments of spheres: if the spheres have a symmetry derived via a stereographic projection from a 4D finite reflection group, then this is a solvable generalization of the familiar problem of a charge inside a spherical cavity. There are 19 three-parametric families of finite groups formed by inversions relative to at most four spheres, each member of each family giving a solvable problem. We solve a sample problem which derives from the reflection group $\mathbf{D}_{4}$ and requires 191 image charges.

中文翻译：

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四维反射群和静电

我们提出了一类新的静电问题，这些问题可以通过添加有限数量的镜像电荷来完全解决。给定一个由多达四个导电接地球体部分包围的腔内某个位置的电荷：如果球体具有通过立体投影从 4D 有限反射群导出的对称性，那么这是一个常见问题的可解推广球腔内的电荷。有 19 个有限群的三参数族，由相对于最多四个球体的反演形成，每个族的每个成员给出一个可解决的问题。我们解决了一个源自反射群 $\mathbf{D}_{4}$ 的示例问题，需要 191 个图像费用。