Elements of the Theory of the Newtonian Potential Function (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1902 Excerpt: ...such functions, V and W, exist, and call their difference -. The function u, then, satisfies condition (2) and is itself equal to zero, or else has its derivative in the direction of the exterior normal equal to zero at every point of the surface. Applying Green's Theorem in the form of Equation 151 to w, we find that the quantity (Dxii)2 + (Dyuy + (2)2w)2, which can never be negative, must be zero at every point within the conductor, so that Dxu, Dvu, and Dzu must vanish and it be a constant throughout the space within the surface. Now at portions of the surface itself, u is zero, hence it must be equal to zero everywhere inside the conductor, and V=W. If by any means, then, we find a function which satisfies the surface conditions and the general space conditions characteristic of the potential function inside a certain conductor carrying a steady current under given surface conditions, this function is itself the potential function. Any surface supposed drawn in a conductor which carries a steady current in such a way that the derivative of the potential function taken normal to this surface is zero shall be called a surface of flow. If a conductor which under given surface conditions carries a steady current be cut in two by means of a surface of flow, and if the two parts be separated while the surface conditions on what was the bounding surface of the old conductor remain the same as before, and the fresh surfaces now abut on an insulating medium, the state of flow at every point inside each part of the conductor will be just the same as before, for the values of V and Dn V on the surface of the new conductors are what they were before separation, and V must have its old values at all inside points. When a conductor is cut in two by a surface of ...