Hyperbolic Cylinder Research Articles

Theory of the photic field

The application of field theory to the calculation of amounts of light has been advocated by a number of investigators. The general feeling, however, seems to be that photic field theory is applicable in so few cases that it is hardly worthy of consideration. A new study of the subject shows that it is of much wider applicability than has been customarily believed. Particularly by the use of the quasipotential, one is able to obtain results very simply that would require troublesome integrations by the classical method. In a few cases, such as the uniform point source, the uniform sphere, and the uniform circular cylinder, the photic field is mathematically identical with the electrostatic field. In these cases, classical potential theory can be applied directly. With most photic fields, however, a scalar potential does not exist and methods must be modified. In all problems having axial symmetry, as well as in other important cases, a quasipotential exists and the procedure is similar to that for a potential field. In still other cases, neither a potential nor a quasipotential exists and one must resort either to a vector potential or to the common photometric methods of calculation. In the foregoing paper, we have attempted to formulate the subject in an exact manner. Theorems have been stated regarding the conditions for the existence of a potential, a quasipotential, and a vector potential in the photic field. Methods of solution are outlined and boundary conditions are formulated. Finally, an example is worked out. Of particular importance are the new Theorems Nos. 7 and 9 on the quasipotential and Nos. 18 and 19 on boundary conditions. Theorem 16 lists the surface sources that produce irrotational fields, while Theorems 21, and 22 indicate general classes of surfaces for which a quasipotential exists. The practical limitation of photic field applications is primarily one of finding a coordinate system. If the luminous surface corresponds to one of the familiar coordinate surfaces (planes, circular cylinders, elliptic cylinders, hyperbolic cylinders, parabolic cylinders, oblate spheroids, paraboloids, etc.), there is ordinarily no difficulty in solving the photic problem with uniform sources. With other problems, difficulty may be encountered, just as in the corresponding electrostatic problem.

On Lattice Points in a Hyperbolic Cylinder

On Lattice Points in a Hyperbolic Cylinder

Ellipsoidal functions and their application to some wave problems

The analysis of wave motion connected with the circular disc and circular aperture is of such great analytical and physical interest, that no excuse is required for a somewhat extended treatment. The theory of wave motion has been very fully developed when the surfaces bounding the medium in which the motion takes place are spheres and circular cones, or cylinders and planes. The functions involved in the analysis are of the hypergeometric type. Next in order of importance is the theory required for the treatment of problems in which the surfaces bounding the medium are :— A. Ellipsoids of revolution and hyperboloids of revolution of one and two sheets ; B. Elliptic and hyperbolic cylinders. The functions involved in case A are functions associated with the ellipsoid of revolution, or spheroid. The functions involved in case B are commonly called MATHIEU functions. The MATHIEU functions and the functions associated with the ellipsoid of revolution are of considerable importance but, unfortunately, of great complexity.

XVII.—Some Hyperspace Harmonic Analysis Problems introducing Extensions of Mathieu's Equation

It is well known that two problems of harmonic analysis in ordinary three-dimensional space can be solved by Mathieu's functions, namely, (a) harmonic analysis for an orthogonal system of elliptic (or hyperbolic) cylinders,(b) harmonic analysis for a system of confocal paraboloïds,

The Solution of Mathteu's Differential Equation

The determination of the harmonic functions of elliptic and hyperbolic cylinders depends on the solution of Mathieu's differential equation. This equation, it has been remarked by Professor Whittaker, is the one which naturally comes up for study after the hypergeometric equation has been disposed of. Its solution presents difficulties which do not arise in connection with the hypergeometric equation or its degenerate cases, and it cannot, I think, be said that any discussion of the equation has yet been given with which the student of analysis can rest content. The treatment given below, though certainly incomplete at some points, seetns to follow the lines along which a thoroughly successful theory may be hoped for.