A C177 TRAINING FLT IS LANDED WITH THE GEAR UP BY THE 2 PLTS ONBOARD AT LVK; CA.

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ON JAN/THU/02; AT APPROX XA00; THE FOLLOWING INCIDENT OCCURRED AT LIVERMORE ARPT. BOTH PLTS FAILED TO CONFIRM GEAR DOWN AND LOCKED. ACFT OPERATED BY PVT PLT AND CFII. THE STUDENT GIVES THE FOLLOWING ACCOUNT. THIS OCCURRED DURING OUR THIRD CIRCUIT IN THE PATTERN AT LVK. BOTH RWYS 7L AND 7R WERE ACTIVE. THE FIRST APCH WAS UNEVENTFUL; WE ENTERED ON A 45 DEGS FROM THE NE AND EXECUTED A TOUCH AND GO ON RWY 7L. WE REMAINED IN THE PATTERN AND WERE CLRED FOR THE OPTION; #4 TO LAND. THE TWR AMENDED THAT CLRNC AND ASKED US TO COME TO A FULL STOP; CLR RWY 7L; TAXI BACK; REMAIN ON HIS FREQ AND HOLD SHORT OF THE RWY. WE WERE TOLD WE WOULD HAVE TO WAIT ABOUT 4 MINS. HERE WE WITNESSED A GAR IN PROGRESS BECAUSE AN ACFT HAD NOT YET CLRED THE ACTIVE. THE TWR CLRED US TO DEPART RWY 7L WITHOUT DELAY. AFTER DEP I HELD THE RWY HEADING. MY INSTRUCTOR ADVISED ME OF TFC BELOW AND TO OUR R DRIFTING TOWARDS US; HE TOOK THE CTL WHEEL AND BANKED L. I RESUMED CTL AND PROCEEDED STRAIGHT FORWARD UNTIL REACHING 800 FT AGL THEN TURNED XWIND. THE TWR INFORMED US OF TFC ENTERING ON A 45 DEGS; I RESPONDED; 'LOOKING FOR TFC.' WE TURNED DOWNWIND IN THE PATTERN. WE WERE CLRED TO LAND; #3. DUE TO THE VOLUME OF TFC; ALL DOWNWIND AND FINAL LEGS WERE EXTREMELY LONG. ON EACH CIRCUIT; APPROX ABEAM THE END OF THE RWY; WE WERE GIVEN OUR ASSIGNMENT FOR LNDG. I VIGILANTLY COUNTED OUT PROCEEDING ACFT SO THAT I COULD IDENT THE ACFT I WOULD FOLLOW. DUE TO THE DISTANCE BTWN EACH ACFT AND SUNNY SKIES; THIS WAS NOT AN EASY TASK. ON LONG FINAL MY INSTRUCTOR DISCUSSED SOFT-FIELD LNDG TECHNIQUE AND HOW HE WANTED ME TO PUT IT ON THE NUMBERS. ON SHORT FINAL WE DISCUSSED GS IN REF TO THE PAPI. WITHOUT THE GEAR LEVER IN THE FULL DOWN POS; WE LANDED THE ACFT GEAR UP. CONTRIBUTING FACTORS: FAILURE TO USE CHKLIST; HIGH VOLUME OF TFC IN THE PATTERN; INSTRUCTION DURING CRITICAL PHASE OF FLT. LNDG GEAR WARNING HORN INAUDIBLE WHILE HEADSETS ARE WORN; LOW TIME IN COMPLEX ACFT; RECENCY OF FLT WITH INSTRUCTOR (LAST TIME FLEW WITH INSTRUCTOR NOV/THU/01; LAST TIME WORKING IN PATTERN WITH INSTRUCTOR OCT/TUE/01). VERY LITTLE PATTERN WORK IN THIS ACFT; LEAKING MAGNETIC COMPASS. NO INJURIES. SUPPLEMENTAL INFO FROM ACN 534762: WE WERE NOW TURNING DOWNWIND AND I FOCUSED ON THE TFC WHILE I FELT VERY COMFORTABLE WITH THE PVT PLT FLYING THE ACFT (I WAS HER INST INSTRUCTOR). I CONTINUED TO RELY ON THE PVT PLT TO RUN THE APCH AS SHE WAS ALREADY CHKED OUT AND SIGNED OFF TO FLY THIS ACFT. I LET MY GUARD DOWN; MAINLY BECAUSE I HAD SUCH CONFIDENCE IN THIS PLT. AND THE PVT PLT ALWAYS DOES THE CHKLIST TWICE ON FINAL. NOW; NO MATTER WHO I FLY WITH I WILL ALWAYS DO MY OWN CHK -- INSTRUCTING OR NOT. FOR PVT PLT FLYING THE APCH THE DISTR BEGAN ON TURNING DOWNWIND; LOOKING FOR THAT TFC COMING IN ON THE 45 DEGS; THAT WOULD BE THE APPROX TIME WHEN THE GEAR WOULD BE LOWERED. I SHOULD HAVE CLRLY SAID 'YOU CONTINUE TO FLY THE ACFT AND LET ME WORRY ABOUT THE TFC' -- I BELIEVE THE PLT WAS ALSO LOOKING FOR TFC AND GOT DISTRACTED. IT WAS THE CONTINUAL DISTR WITH VERY BUSY TFC THAT LED TO THE GEAR UP LNDG.