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Abstract
In the Standard Model, the experimental fact that the quark weak interaction eigenstates are not the mass eigenstates permits a set of otherwise forbidden processes, including flavour changing neutral currents (FCNC) and CP violation. This is the phenomenology of the quark-mixing matrix, $V\sb{\rm CKM}$. While it, together with the fact that quarks have differing masses, allows these processes to occur, they are by no means encouraged. This strong Standard Model suppression allows relatively significant contributions from the host of theories of physics at higher energy scales. Such contributions may not make any single processes inconsistent with the Standard Model, but their different contributions to processes with related $V\sb{\rm CKM}$ dependencies may leave a picture inconsistent with the current theory. Exploring $V\sb{\rm CKM}$ is an important part of the near-term program of particle physics, especially at the two new B-factories under construction. A measurement of the branching ratio of the rare kaon decay $K\sp+\to\pi\sp+\nu\overline\nu$ is a theoretically clean measurement of the product of $V\sb{\rm CKM}$ elements $V\sbsp{ts}{*}V\sb{td}$, currently the poorest known elements of the mixing matrix, constrained primarily by the unitarity of $V\sb{\rm CKM}$ demanded by the theory. The dependence of $K\sp+\to\pi\sp+\nu\overline\nu$ on various types of new physics is complementary to that of the measurements that will be made at the B-factories, so $K\sp+\to\pi\sp+\nu\overline\nu$ will remain of interest into the future. Experiment 787 at Brookhaven National Lab is a dedicated search for $K\sp+\to\pi\sp+\nu\overline\nu$, expected in the Standard Model to have a branching fraction of (0.3-3) $\times$ 10$\sp{-10}$; the range is due to the uncertainties in the $V\sb{\rm CKM}$ parameters themselves. This thesis is a complete analysis of our existing data set, yielding no candidate events. A 90% CL upper limit of 2.43 $\times$ 10$\sp{-9}$ on the $K\sp+\to\pi\sp+\nu\overline\nu$ branching fraction is inferred. A simultaneous search for the processes $K\sp+\to\pi\sp+X\sp0$, where $X\sp0$ is any massless, weakly interacting particle, results in a 90% CL upper limit on the branching fraction of 5.18 $\times$ 10$\sp{-10}$.
