What was found
DURING FINAL APPROACH, FLAPS WERE SELECTED TO FULL DOWN FROM APPROACH POSITION. PILOT HEARD A LOUD BANG AND AIRCRAFT 'SETTLED'. A GO-AROUND WAS PERFORMED. PILOT NOTICED THE FLAP POSITION INDICATOR SHOWED FLAPS UP WITH SELECTOR IN THE FULL DOWN POSITION. A NO-FLAP LANDING WAS PERFORMED. DURING TAXI, PILOT NOTED FLAP INDICATOR MOVING TOWARD FULL DOWN POSITION. FLAPS WERE FULLY DOWN AT ENGINE SHUT-DOWN. FLAP GEARBOXES WERE REMOVED AND DISASSEMBLED. FOUND THE SMALL GEAR TEETH SHEARED OFF OF BOTH TRANSFER GEARS. BOTH GEARBOXES WERE OVERHAULED AND REINSTALLED. PILOT ALSO NOTED FLAP MOTORS CONTINUED TO RUN AFTER GEARBOX FAILURE UNTIL CIRCUIT BREAKER WAS OPENED.