Exact Expressions for the Gravitational Attraction of a Circular Lamina at All Points of Space and of a Right Circular Vertical Cylinder at Points External to It

The gravitational attraction of a circular lamina and a right circular vertical cylinder at any axial point can be found by elementary methods. The attraction at a point P off the axis is usually estimated as follows. The attraction of a lamina, with the axis through P, touching the given lamina externally is subtracted from the attraction of a coaxial lamina which encloses the given lamina. The result is multiplied by the ratio of the area of the given lamina to the area of the annulus. The attraction of the cylinder is estimated by a closely analogous method. The answer is, however, valid only if P is a distant point. As a step towards calculating exactly the attraction of the cylinder at all points of the space external to it, the corresponding problem for the lamina is dealt with first. It is shown that the anomaly can be expressed as an infinite series in even powers of the horizontal distance of P. The coefficients of the powers consist of exact algebraic expressions containing simple terms of the form (1 +χ2)−n−1/2. These coefficients are determined by exactly summing certain infinite series involving the derivatives of the Legendre polynomials of odd order. The method of summation is believed to be of interest also in other problems where Legendre polynomials are encountered. The attraction of the cylinder is calculated by appropriately integrating the expression for the attraction of a lamina. Numerical calculations show that the approximate method everywhere underestimates the gravity anomaly of the lamina as well as of the cylinder. The error at distances one to two times the radius is upto 20 % for the lamina and 8.5 % for an infinitely long cylinder.