BE33 pilot reports NMAC with Ercoupe during arrival to an uncontrolled airport. The pilots involved were in radio communication but failed to establish positions.

Narrative

I canceled my IFR flight plan with Approach Control; which they granted; and I was then instructed to squawk 1200. Approach Control did not advise; and I forgot to ask; if there was any traffic in ZZZ's pattern. Normally Approach Control does advise of possible traffic in the pattern; and I normally request an advisory if they fail to make one voluntarily. I was still high due to Approach Control's late authorization for lower. My advisory radio call into ZZZ was made immediately on CTAF frequency when I was approximately nine miles out. The call was as follows: 'ZZZ traffic; Bonanza X is nine miles to the south; inbound ZZZ...full stop. We will be crossing mid-field at 2;000 FT entering the left downwind for Runway XX; ZZZ traffic.' My radio call went without acknowledgment...complete radio silence.At approximately six miles out a second advisory radio call was made by me into ZZZ's CTAF frequency. The radio call was identical to the first radio call; except for the distance. 'ZZZ traffic; Bonanza X is six miles to the south; inbound ZZZ...full stop. We will be crossing mid-field at 2;000 FT entering the left downwind for Runway XX; ZZZ traffic.' Once again; my radio call went unanswered...complete radio silence occupied my cockpit. Our approximate altitude was somewhere around 4;500 FT at the time; with the power setting remaining the same. However; the configuration of my aircraft was changed; as we dropped the gear. Once the gear was down and locked I put in about 10 degrees of flaps.My third radio position and advisory call came when I was about three miles out. The call was pretty much verbatim to the two previous advisory calls. This time; a radio call was received from Aircraft Y; 'Aircraft landing ZZZ; where are you?' He gave no call sign. My reply was; Aircraft X is three miles to the south. Where are you?' Aircraft Y replies; 'I'm left downwind for Runway XX; you're okay.' Again there was no call sign. Because we were fast approaching ZZZ; I replied; 'Aircraft X will fly over the departure end of ten; and will then fly to the northeast.' No response was received from Aircraft Y. I then retracted both gear and flaps and headed on the announced course of a northeast heading. I was now about a mile and half off of the departure end of ten. My eyes peering down the left downwind for the unidentified aircraft. No contact. I had climbed to around 3;000 FT and was headed northeast on a heading; I'm guessing close to 060 to 065 degrees; and all along looking towards the downwind base; and final legs for Aircraft Y. I continued on this heading and altitude until we were close to 4 miles or so from the approach end of Runway YY. At this point; I make another advisory/position call. 'ZZZ traffic; Aircraft X is turning inbound for ZZZ; We will be entering on a long left downwind for XX; ZZZ.' Again; there was no response. I began a right descending turn and was in perfect position for a long left downwind for XX. After completing my turn; I continued to scan for the traffic...concentrating on the west end of the airport; since Aircraft Y initially reported he was 'left downwind for Runway XX.' Aircraft Y should be either on final or on the ground. Seconds later; my wife let out a startled cry of surprise and gasped! I looked over to her quickly and then noticed the Ercoupe; Aircraft Y; passing under our belly headed in a northeasterly direction 25 to 30 FT away! We took no evasive action as the near-miss had already occurred. I'm not sure if Aircraft Y took any kind of evasive action prior to the near-miss or that he even saw us. This is because; again; we did not see the Ercoupe until it had passed underneath us. I'm guessing we were about two miles; maybe two and half miles from the approach end of YY when the event occurred. Anyway; for a stunned few seconds I continued to fly on my left downwind heading; now realizing of what just occurred I made a stern radio call to the Ercoupe. 'Ercoupe; continue to fly your present heading and I don't want to see your face until I've landed.' Of course; Aircraft Y made no acknowledgment of my scolding call.Minutes later; we landed safely on Runway XX and rolled to the intersection of Runways ZZ/WW. I radioed that we were back-taxing Runway ZZ; and radioed again when we were cleared Runways ZZ/WW. We taxied to parking; all along with radios live to listen for the perpetrator. Nothing! Just before shutdown; I saw the Ercoupe in a mid left downwind for 10. He radioed his position. I don't remember him calling any other position before his touch and go landing. The Ercoupe made its touch and go about the ZZ/WW intersection; climbed to about 200-250 FT and made a left climb-out. He did not state his intentions to stay in the pattern. At mid-downwind for ten he radioed his position. I shut the power down to our aircraft and we began the process of stowing our airplane. The Ercoupe pilot landed; taxied to parking and was long gone before we had stowed our luggage in our car.