Passing midway between Green Island, which is about twenty feet high, encircled with a coral reef, and Cape Grafton, we steered North-West 1/2 North for a shoal on which Her Majesty's Ship Imogene grounded; and at noon, were exactly on the spot, in latitude 16 degrees 24 1/4 minutes South by observations and bearings of the land, Low Isles being West-North-West four miles. Here we found sixteen fathoms, not having had less than seventeen since the morning. There was no appearance of any such reef nearer than that laid down by Lieutenant Roe, bearing east from the above-mentioned Low Isles and under which Her Majesty's Ship Tamar anchored. It must therefore have been on the North-West part of this reef that the Imogene struck, and the south part must be the reef laid down in the chart as having been seen by her to the southward, which accounts for our not seeing it from the Beagle. We passed through several patches of discoloured water, caused by washings from reefs to windward, which are very deceptive. At sunset the anchor was dropped in thirteen fathoms, for the first time since leaving Port Stephens. The south point of Weary Bay bore West-North-West three miles, and Cape Tribulation South by East six miles. Near the middle of the former, I noticed a patch of discoloured water, which has since been found by a merchant vessel to be a shoal. HOPE ISLANDS. The land over the latter place is very high, presenting several singular peaks, one more prominent than the rest, in the shape of a finger. That over Trinity Bay, which we were the greater part of the day crossing, is also of great altitude. In its south corner we noticed the river-like opening spoken of by Captain King, lying in the rear of some remarkable peaks. We had been informed by him, that the greater part of the coast between Weary Bay and Endeavour River, including the Hope Islands, had been altered from his original survey, a tracing of which he had furnished us with previous to leaving Sydney. The few bearings we obtained while at anchor, induced us to consider it correct, a fact we further proved during the early part of the next day's run, as the course steered from our anchorage North by West 1/2 West, carried us a little more than a mile west of the Hope Islands. Had their assigned position in the chart been correct, our course would have led us right over the western isle. On detecting this error, we found it necessary to re-survey this part of the coast, and it affords me much pleasure, after so doing, to be able to bear testimony to the extreme correctness of Captain King's original chart above alluded to. Soon after passing the Hope Islands, we saw the reef where Cook's vessel had so miraculous an escape, after grinding on the rocks for 23 hours, as graphically described in his voyages. It is called Endeavour Reef, from this circumstance. CAPE BEDFORD.