THE LOCATION OF THE ZEROS OF THE DERIVATIVE OF A RATIONAL FUNCTION, REVISITED,
Abstract
From the general standpoint of circle geometry and linear transformations of the plane, the following theorem is formulated: Let there be given a (closed) circular region C and two fixed points Z and z exterior to C. Let a number of particles have C as their locus with the requirement that their center of gravity with respect to Z shall be the inverse of Z in the boundary of C. Then the locus of the center of gravity of these particles with respect to z is the closed region not containing Z bounded by that circle of the coaxial family determined by the boundary of C and the null-circle Z which passes through the harmonic conjugate of z with respect to the intersections with the boundary of C of the circle through z of the conjugate coaxal family. This theorem is established for the special case Z = infinity. This is the most important special case, for if a set of points (weighted particles) in the plane is given, their center of gravity is easily found, as is a disc with the center of gravity as center containing the given points. Nevertheless some attention is devoted to the special case of this theorem when Z is the center of a disc whose closed exterior is C. Most of the conclusions deal with lemniscates and curves of higher degree. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1940
- Accession Number
- AD0428343
Entities
People
- J. L. Walsh
Organizations
- Harvard University