Confocal Ellipses Research Articles

The Use of Current Sheets in the Design of Magnets to give Bounded Fields of Required Form, free from Edge Distortion

The use of current sheets to produce static magnetic fields of required form, with freedom from edge or fringing distortion and with no stray flux, is discussed. Two types of magnet winding are considered. The first, or field-terminating type, encloses a tube of force and serves as a sharp boundary to the field. The second, or iron-shrouded type, is wound against the inner surface of an iron shroud and, with a suitable distribution of turns, can generate any required shape of field in the iron enclosure. In both types stray field can be avoided if the iron return-path is infinitely permeable, or if it is covered with an `anti-reluctance' winding. A discussion is given of (a) the uniform field, (b) the field whose lines of force are coaxial circles, (c) the synchrotron field, and (d) the prolate spheroidal field, whose lines are confocal ellipses

The Moment of Inertia and the Period of the Rocking of an Elliptic Band

The radius of gyration of a thin, elliptical band of uniform line density is given (i) by direct evaluation, (ii) from the point of view that the band represents the interval between adjacent ellipses of semiaxes a and b, and (iii) by taking the band between two confocal ellipses. Values of the radii of gyration for several elliptic bands are computed from the three points of view. Experiments in which elliptic bands of brass and glass were rocked on glass plates and on suspended knife-edges gave agreement within 5 percent of the computed radii of gyration.

XII.—Green's Function for an Ellipse, and its Application to the Motion of a Point Vortex

SummaryGreen's function for a plane outside an ellipse and for a plane between two confocal ellipses are evaluated.The motion at a point vortex outside an ellipse is found.

Deriven Orthogonal Systems from Confocal Ellipses and Hyperbolas with the Aid of N=th Root Process.

Deriven Orthogonal Systems from Confocal Ellipses and Hyperbolas with the Aid of N=th Root Process.

Some properties or confocal ellipses and their application to mechanism

Some properties or confocal ellipses and their application to mechanism

VI.—Note on Confocal Conic Sections

A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, was inserted in the twenty-third volume of the Transactions of the Royal Society. Some of the conclusions of that paper can, however, be obtained more simply, as I will now proceed to show.I will, in the first place, resume the problem—“To find the intersection of a confocal ellipse and hyperbola.”Since the curves have the same foci, and therefore the same centre, let the distance between the centre and focus be called unity, since it is the same for both curves. Let a, b, be the axes of the ellipse, A, B, those of the hyperbola. Then we have 1 = a2 − b2 = A2 + B2, which equation expresses the condition of confocality.