Abstract: A novel approach to the numerical analysis of the dynamic system of two spiraling black holes is developed using the initial value formalism of general relativity. As the two black holes spiral in on each other they will lose energy and angular momentum in the form of gravitational radiation. When the amplitude of this gravitational radiation is small enough, the space-time for the two orbiting black holes can be approximated by a geometry which is unchanging as seen by an observer in a frame of reference co-rotating with the two black holes. Then a time-like Killing vector field is assumed to exist over a finite region of the space-time geometry. A variational principle is found for the total mass of the binary system based on the Hamiltonian of general relativity and is used to study the dynamics and stability of the close orbits. The emission of gravitational waves within the context of the quadrupole moment approximation is used to determine the secular evolution of the system. For black holes of equal mass, approximately 3% of the their initial mass, as deter- mined when the two black holes are at rest at large separations, is emitted as gravitational radiation with frequencies less than the quadrupole normal mode frequency of the final coalesced Kerr black hole. Dissertation Discovery Company and University of Florida are dedicated to making scholarly works more discoverable and accessible throughout the world. This dissertation, "The Spiralling Binary System of Black Holes" by James Kent Blackburn, was obtained from University of Florida and is being sold with permission from the author. A digital copy of this work may also be found in the university's institutional repository, IR@UF. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation.