均匀长椭球体的引力互能。共线情况
Mutual gravitational energy of homogeneous prolate spheroids. Collinear case
摘要：
建立并求解了对称轴在同一直线上的两个均匀长椭球体系统的相互引力能$W_{mut}$问题。采用等重力单元法，将三维球体的外部势用一维非均匀焦棒的势表示。问题的解归结为一个杆的势能在第二个杆的线段上的积分。结果表明，利用初等函数可以得到两个长椭球的有限解析表达式$W_{mut}$。求出了椭球体间的引力，并将函数$W_{mut}$用椭球体偏心率的幂级数表示。所获得的结果的可能的应用进行了讨论。
英文摘要：
The problem of mutual gravitational energy $W_{mut}$ for a system of two homogeneous prolate spheroids, whose symmetry axes are on the same line, is set and solved. The method of equigravitating elements is applied, where the external potentials of three-dimensional spheroids are represented by the potentials of one-dimensional inhomogeneous focal rods. The solution of the problem is reduced to the integration of the potential of one rod over the segment of the second rod. As a result, the expression $W_{mut}$ for two prolate spheroids can be obtained in a finite analytic form through elementary functions. The force of attraction between the spheroids is found. The function $W_{mut}$ is also represented by a power series in eccentricity of the spheroids. Possible applications of the obtained results are discussed.
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