  We develop a stream function approach for the horizontal Hele-Shaw,
Saffman-Taylor finger. The model yields a nonlinear time-dependent differential
equation. The finger widths derived from the equation are
$1>\lambda>\frac{1}{\sqrt{5}}$, in units of half the width of the Hele-Shaw
cell, in accordance with observation. The equation contains the correct
dispersion relation for the creation of the finger instability. In an
accompanying paper the stationary solutions of the equation are found
numerically.
