The nearest integer continued fractions of Hurwitz, Minnegerode (NICF-H) and in Perron's book Die Lehre von den Kettenbrüchen (NICF-P) are closely related. Midpoint criteria for solving Pell's equation x² - Dy² =   ± 1 in terms of the NICF-H expansion of √D were derived by H. C. Williams using singular continued fractions. We derive these criteria without the use of singular continued fractions. We use an algorithm for converting the regular continued fraction expansion of √D to its NICF-P expansion.