2 B737'S ON SHORT FINAL REACTED TO TCASII RA'S FOLLOWING ONE ACFT'S DEV FROM ITS RESPECTIVE FINAL APCH COURSE.

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I WAS TRAINING A CTLR ON THE ARR POS FOR DAL. WE WERE UTILIZING VISUAL APCHS TO RWY 13L&R. THE 2 ACFT WERE APCHING THE ARPT FROM DIFFERENT DIRECTIONS. ACR #1 WAS ON VECTORS; FROM OVER DFW ON A R BASE AT 4000 FT; INDICATING 240 KTS. ACR #2 WAS ON VECTORS; OVER ADS (5 MI NNE OF DAL) AT 3000 FT; INDICATING 240 KTS. ACR #2 WAS CLRED TO 2000 FT; ASSIGNED HDG 240 DEGS; AND ASKED TO RPT THE ARPT IN SIGHT. THE PLT OF ACR #2 ACKNOWLEDGED. ACR #1 WAS CLRED TO JOIN THE LOC FOR RWY 13R. AT THIS TIME; THESE ACFT WERE APPROX 6 MI APART LATERALLY AND 1000 FT VERTLY. ACR #1 TURNED STRAIGHT-IN TO RWY 13R AND APPEARED TO BE SLOWING BECAUSE GND SPD INDICATED 210 KTS. ACR #2 APPEARED TO BE DSNDING SLOWLY AND STILL INDICATED 240 KTS. ACR #2 WAS INSTRUCTED TO TURN TO HDG 160 DEGS; FOR A 30 DEG INTERCEPT (AS IN ACCORDANCE WITH FAA DIRECTIVES) AND ASKED TO RPT THE ARPT IN SIGHT. ACR #1; APPROX 8 MI STRAIGHT-IN; RPTED THE ARPT IN SIGHT AND WAS CLRED FOR A VISUAL APCH TO RWY 13R. ACR #2 RPTED THE ARPT IN SIGHT; WAS CLRED FOR VISUAL APCH TO RWY 13L; AND INSTRUCTED TO JOIN FINAL FROM THE 160 DEG HDG. AT THIS TIME THESE ACFT WERE APPROX 4 MI APART LATERALLY AND 500 FT VERTLY. ACR #1 APPEARED TO BE ON A NORMAL APCH TO RWY 13R. ACR #2 HAD AN INDICATED GND SPD OF 230 KIAS AND APPEARED TO BE VERY SLOW IN COMING AROUND TO HDG 160 DEGS. TFC WAS EXCHANGED. ACR #1 RPTED ACR #2 IN SIGHT BUT; AT THAT TIME; WASN'T INSTRUCTED TO MAINTAIN VISUAL SEPARATION. SUBSEQUENTLY; THESE ACFT CONVERGED. ACR #2 WENT THROUGH BOTH FINAL APCH COURSES AND ENDED UP BELOW AND IN FRONT OF ACR #1 APPROX 3 MI FROM THE ARPT. ACR #1 WAS ASKED IF HE STILL SAW ACR #2. ACR #1 SAID HE STILL HAD ACR #2 IN SIGHT AND HAD TO DO A TCASII CLB. ACR #1'S APCH CLRNC WAS CANCELED; HE WAS CLRED TO MAINTAIN 3000 FT AND INSTRUCTED TO MAINTAIN VISUAL SEPARATION FROM ACR #2. AT THIS TIME; ACR #2 APPEARED TO BE TRACKING IN ON THE FINAL APCH COURSE TO RWY 13R. ACR #2 WAS ASKED IF HE WAS TURNING TO RWY 13L AND SAID HE WAS; THEN MUMBLED SOMETHING UNINTELLIGIBLE BEFORE BEING TOLD TO CONTACT THE TWR. ACR #2 SUBSEQUENTLY DID A GAR TOO. I BELIEVE THIS EVENT COULD'VE BEEN AVOIDED IF ACR #2 WAS AT A MORE MANAGEABLE SPD; EITHER PLT INITIATED OR CTLR ASSIGNED. SUPPLEMENTAL INFO FROM ACN 635095: UPON ARR AT LOVE FIELD; WE WERE TO 2 MI S OF ADDISON ON A L DOWNWIND RWY 13L LOVE FIELD. WE WERE CLRED TO 2000 FT AND TOLD TO CALL THE FIELD WHEN IN SIGHT. WE WERE GIVEN A 270 DEG HDG; THEN A 210 DEG HDG AT WHICH TIME WE CALLED THE FIELD. WE WERE THEN GIVEN 160 DEG HDG AND CLRED THE VISUAL. BY THE TIME WE ROLLED ON A 160 DEG HDG; WE WERE GOING THROUGH THE RWY 13L LOC AT WHICH TIME WE WERE LOOKING FOR TFC FROM A TA. THE FO VISUALLY SAW THE TFC AND MANEUVERED THE ACFT TO STAY CLR TO COMPLY WITH THE RA. (THE TA WAS THE FIRST INDICATION OF TFC.) WHEN WE WERE COMFORTABLE THAT ACR #1 WAS A SAFE DISTANCE; WE WENT AROUND AND LANDED WITH NO OTHER PROBS. SUPPLEMENTAL INFO FROM ACN 635102: FO. I WAS THE PLT MONITORING. ON FINAL APCH TO RWY 13R ABOUT 7-8 MI FROM RWY; STILL TRYING TO WORK DOWN TO NORMAL DSCNT PROFILE; REGIONAL APCH WARNED US THAT ANOTHER COMPANY AIRPLANE WAS TURNING L BASE TO THE PARALLEL RWY (13L). ABOUT THE SAME TIME THE TCASII SYS WARNED US OF 'TFC; TFC.' WE VISUALLY ACQUIRED ACR #2 ON BASE TURN AND CONTINUED. ACR #2 SEEMED TO OVERSHOOT BASE TURN AND INCREASED BANK TO AVOID ENCROACHING INTO OUR AIRSPACE. HOWEVER; THE TCASII AT THAT TIME GAVE US AN RA AND DIRECTED US TO CLB AT 2000 FPM. INITIATED GAR AND TCASII RA PROCS. APCH CTL INITIALLY GAVE US A 3000 FT MSL LEVELOFF ALT AND A L TURN TO 060 DEGS. APPARENTLY; ACR #2 ALSO WENT AROUND AND THEY; TOO; CLBED OUT TO 3000 FT MSL. SEEING THE APCHING CONFLICT; REGIONAL APCH GAVE US AN IMMEDIATE CLB TO 4000 FT MSL. WE DID HAVE COMPANY ACFT IN SIGHT AND DID NOT FEEL THAT WE WERE IN AN UNSAFE POS. ONCE THE TCASII DIRECTED AN IMMEDIATE CLB; WE FOLLOWED THE RA CLB PARAMETERS AND CONFIRMED THE POS OF OUR COMPANY ACFT.