The calculation of magnetic attraction by the aid of magnetic figures

The present paper treats of the following matters: 1. Simplification of the physical formula for magnetic attraction 2. Definition of magnetic reluctance when the magnetic field is bounded by non-equipotential surfaces 3. Calculation of the virtual variation in magnetic reluctance due to any displacement of the limiting surfaces 4. Demonstration of the theorem that the magnetic attraction between two ferromagnetic bodies depends only upon their common magnetic fluxes φ and upon the virtual gradient of its air-reluctances R
o
, and that the formula
$F_{l}={\phi^2\over 8\pi} {\lambda R_{o}\over \lambda_{l}}$
which gives the attraction along any given direction l is as general and as exact as the formulas of physics 5. Direct deduction of magnetic attraction from the lines of magnetic flux depicted by a magnetic figure without resorting to the components of the magnetic field. The method used is developed by the application of the well-known principles of the potential energy function to the magnetic flux in the air-gap. Paths for the magnetic flux are established across the air-gap between the two ferromagnetic surfaces bounding the air-gap, by decomposing the magnetic flux into elemental tubes of magnetic force, the envelopes of which enclose spaces in which the flux is constant. The element of boundary-surface intersected by each elemental tube at the two boundary-surfaces encloses an equal number lines of magnetic flux, irrespective of the magnetic density at these points. By replacing the element of boundary-surface by its geometrical projection on a plane normal to the axis of the elemental tube, an equivalent equipotential surface is obtained at each end of the elemental tube. Any non-equipotential surface bounding an air-gap can thus be replaced by an equivalent equipotential surface composed of an aggregation of elemental equipotential surfaces which produce denticulations in the contour of the boundary-surface. It becomes possible, in this way, to evaluate magnetic reluctance and magnetic attraction by reference to summations of elemental magnetic tubes and without the necessity of considering directly the magnetic density of the magnetic field at any point. The potential of each elemental tube of magnetic force depends only upon the potentials at its ends, at the boundary-surfaces, it being entirely independent of the path followed by the tube in traversing the air-gap.