DC9 REQUIRED TO MAKE 2 TCASII MANEUVERS FOR 2 B737'S WHEN XFERRED FROM ONE ZME CTLR TO ANOTHER.

Narrative

WE EXPERIENCED RA #1 ACCORDING TO THE ABOVE DATA. WE ALSO HAD ANOTHER RA IN WHICH EVASIVE ACTION WAS TAKEN. APPROX 30-60 SECONDS LATER THE CAPT STARTED THE RECOVERY FROM RA #1. THIS (RA #2) OCCURRED IN THE DSCNT (FROM FL291 PEAK ALT FROM RA #1) AT FL284. THIS EVASIVE ACTION REQUIRED THE CAPT TO CLB TO FL295. THE RA'S OCCURRED ON 118.27. WE HAD JUST BEEN TURNED OVER FROM 124.27; ZME. THE FIRST ENCOUNTER HAPPENED WHILE WE WERE IN CRUISE FLT. AFTER THE EVASIVE ACTION; I (THE FO) LOOKED FOR THE TFC AND SAW A B737 OUT MY R SIDE WINDOW. THIS DISTANCE WAS NOT ONLY CLOSE; BUT FRIGHTENING IN THAT THE B737 WAS NOT OR DIDN'T APPEAR TO BE IN EVASIVE ACTION. THE B737 WAS CLBING RIGHT THROUGH OUR ALT AS IF THEY WERE COMPLETELY UNAWARE OF OUR POS. AFTER THE FIRST RA AND WE STARTED THE RECOVERY TO FL280; WE ENCOUNTERED ANOTHER RA AT FL284 IN THE DSCNT. THE CAPT (PF) RESPONDED AND BEGAN EVASIVE ACTION IN RESPONSE TO THE TCASII INSTRUCTIONS. THIS TIME WE ENDED UP 300 YARDS FROM ANOTHER B737. TOO CLOSE FOR ANYONE'S COMFORT! I BELIEVE THE PROB HERE WAS POOR COM BTWN CTLR HANDLING US AND THE B737. POOR COM BTWN THE 2 CTLRS HANDING US OFF TO THE NEXT.