The Fourier series for the charge density on a hollow tube of finite length is found in such a way that the Fourier coefficients are the unknowns of a system of linear equations. A machine method is used to solve the set of equations in finite order. The leading coefficient is the most accurately known at any order, and solely determines the capacitance of the tube. Capacitances are determined for various ratios of half-length to radius to an accuracy of better than 0.02%. The technique of solving the integral equation for charge density is useful in other physical problems.