Gauss' Theorem

If any closed surface S, drawn in a field of force, be divided up into a large number of surface elements, and if each one of these elements be multiplied by the component in the direction of the interior normal of the force of attraction at a point of the element, and if these products be added together, the limit of the sum thus obtained, as the elements are made smaller and smaller, is called the surface integral of normal attraction over S. This theorem a erts that the surface integral of normal attraction over any closed surface whatever is equal to 47 times the quantity of matter it incloses; or fNds=fDVds=47M; where N is the normal component of the force, and V the potential function at the point. Clearly, if all the attracting matter is without the closed surface, the integral is equal to zero.