A new algorithm for analyzing ring diagrams from helioseismic data is presented. This method is applied to two months of Solar and Heliospheric Observatory/Michelson Doppler Imager data from 1996 May 24 through July 24 to study flows and to measure the ratio of the horizontal displacement to the vertical displacement for the oscillations. We find that (1) the rotation rate agrees well with that measured from global modes, extending the measurements closer to the surface; (2) there is a 20 m s-1 poleward meridional flow; (3) there are medium-scale velocity features persisting for several days; (4) the radial gradient of the near surface rotation rate decreases with latitude; and (5) meridional flow decreases with depth. The measured horizontal-to-vertical displacement ratio is in agreement with that expected from theoretical considerations.