A B737-300 ON DEP FROM DCA PENETRATED P56 FOLLOWING DUAL FMC FAILURES.

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DEPARTING DCA AS FLT X; THE B757 DEPARTING BEFORE US HIT A FOX ON THE RWY SO WE WERE CLRED INTO POS RWY 1 AND HELD FOR SEVERAL MINS WHILE VEHICLES SWEPT FOR DEBRIS. SEVERAL ACFT ON APCH WERE SENT AROUND AND TOLD TO INTERCEPT THE DCA 328 DEG RADIAL OUTBOUND. WE WERE SET UP TO FLY THE FMS GUIDANCE PROC AND HAD BRIEFED AND QUICK-ALIGNED THE IRS'S IN ACCORDANCE WITH COMPANY PROCS. WHEN CLRED FOR TKOF; THE CAPT FLEW THE RNAV DEP; STARTING THE TURN UP-RIVER TOWARDS TEDIE AT 400 FT AS WE ENTERED THE OVCST. FOR THE NEXT FEW SECONDS; I WAS OCCUPIED AS I CUT MY THUMB ON THE GLARESHIELD WHEN RAISING THE GEAR. IT WAS BLEEDING PROFUSELY AND BY THE TIME I GRABBED A NAPKIN AND LOOKED BACK UP; I SAW WE WERE HEADING APPROX 330 DEGS BUT THE FLT DIRECTORS WERE COMMANDING A STEEP R TURN. AT THIS TIME POTOMAC AP COMMANDED AN IMMEDIATE R TURN TO 050 DEGS. AT THE TIME WE ASSUMED THIS WAS FOR SPACING BEHIND THE GA ACFT SENT UP THE 320 DEG RADIAL WHILE WE WERE WAITING ON THE RWY. HOWEVER; DURING THE NEXT COUPLE OF VECTORS WE RECEIVED THE FMS ALERT MESSAGE 'VERIFY POS.' WHEN POTOMAC APCH CLRED US DIRECT GINYA INTXN; WE WERE UNABLE DUE TO FMC FAILURE (FMC ANNUNCIATED IN MIDDLE OF BLANK SCREEN). AT THIS POINT; WE NOTIFIED ATC OF OUR EQUIP FAILURE; AND REQUESTED TO CONTINUE FLT AS '/A' (VOR ONLY). CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR'S CALLBACK INCLUDED HIS RESEARCH INTO THE MECHANICAL FAILURE RESPONSIBLE FOR THE DEV. ONE OF THE 2 FMC'S EXPERIENCED A HARDWARE FAILURE AND BEGAN SENDING INCORRECT INFO TO THE OPPOSITE FMC. THE GOOD FMC SHUT DOWN BECAUSE THE CONFLICTING INFO CREATED UNCERTAINTY IN THE COMPUTER LOGIC ABOUT THE VALIDITY OF THE GOOD FMC. THE CREW INITIALLY THOUGHT THE VECTORS WERE FOR TFC SEPARATION FROM PRECEDING ACFT EXECUTING GAR'S FOLLOWING THE RWY CLOSURE FOR A DEAD FOX ON THE RWY. IN FACT THE VECTORS WERE TO KEEP THEM AWAY FROM THE NAVAL OBSERVATORY (VICE PRESIDENT'S RESIDENCE) AFTER THEY NARROWLY MISSED THE PRESIDENT'S QUARTERS (P56) WHILE FOLLOWING FAULTY FMC HEADING GUIDANCE. THE CREW HAS NOT HAD ENFORCEMENT ACTION INITIATED AGAINST THEM. THE ACR POI AND PLT UNION REVIEWED THE INCIDENT AND THE RPTR BELIEVES THE CREW HAS BEEN CLRED OF A VIOLATION BECAUSE OF THE NAV HARDWARE FAILURE.