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This web page is a report on the research done by the author in the summer of 1999. In general, it discusses families of continued fractions, and specifically, it presents a method for proving that the partial quotients of a family of continued fractions come from the square root of a given function. The type of families discussed are those that have the form $\sqrt{d} =
\sqrt{f(x_1, x_2, \ldots, x_n)}$, where the primitive period length of the continued fraction for $\sqrt{d}$ grows linearly with xi, for some $1 \leq i \leq n$, or linearly with respect to the logarithm of $\sqrt{d}$. The report will culminate with the investigation of two specific families, on which a method of proof can be applied that verifies certain properties of the families. Although there is such an emphasis on continued fractions, no familiarity with them is assumed. For a more formal report on this research, see  [5].