A ring-diagram analysis technique has been applied to a region of the Sun of about 15 deg^2 area close to the disk center at dates chosen for several solar rotations in 1996: April 5, May 1, May 28, June 27, July 23, August 20, August 21, and September 18. The horizontal velocity flows inferred directly beneath the solar surface have thus been obtained for the same region of the Sun at the given dates. The data have been obtained at the Observatorio del Teide and the Tashkent stations with the Taiwan Oscillation Network (TON), consisting of 1080 x 1080 pixel Ca II K-line intensity images. The stability of the results, the stability of the method, and the possible temporal evolution of the subsurface flows have been studied from the results. A rotation curve for the differential radial solar rotation down to the first 27 Mm of depth at the equator is also suggested and qualitatively compared to results obtained by other authors.