Apply a moderate amount of mineral oil to a clean cloth in an open area with considerable ventilation. Have a box fan nearby if extra ventilation is needed.
Wipe down the Morse taper thoroughly to remove any grease, grime or dirt that may be on the taper. Also clean out the drill spindle as best you can with a cloth coated in the mineral spirits. Use a screwdriver to guide the cloth into the spindle if necessary.
Look up into the drill spindle, and notice the direction of the slot in the drive mechanism. Align the rectangular tang on the end of the taper with the slot in the spindle, and insert the taper into the spindle. Rotate the taper as necessary to ensure tang alignment inside the slot.
Hold the Morse taper with one hand and tap the end of the taper securely with a wooden or rubber mallet. Do not hit the taper directly with a metal hammer, as you will likely damage the taper.
Retract the jaws of the drill press chuck so that they are fully housed within the body of the chuck. Place the back end of the chuck over the exposed (bottom) end of the installed taper, and tap the chuck into place with the rubber or wooden mallet.
Insert a drill bit into the jaws of the chuck and tighten the chuck with a chuck key. Test the drill press by drilling a couple of holes to ensure that the taper and chuck are securely installed.