The integrals referred to occur in connection with applications of the Method of Electric Images. A curious special case was given to the Society in July 1875, but was not inserted in the “Proceedings.” It depended on the fact that the image of a sphere, whose surface density is inversely as the cube of the distance from a point, is another sphere with a similar law of distribution. But any desired number of integrals, whose values can be at once assigned, may be obtained by various applications of the following process.