CASTELLAMARE, NAPLES, Aug. 8.

The court-martial on Lieutenant Bedingfield, of the Hecate, ended on Saturday afternoon, when, after four hours' deliberation by the Court, the sentence was, that he should be dismissed his ship and put back one year on the Navy List. The charge which called forth this sentence was having gone on shore, whilst commanding-officer of the ship, against the Commander's written orders; but which order he (the prisoner) thought had been cancelled. Lieutenant Bedingfield has since been discharged into the Vengeance, there to await a passage to England, or to proceed home at his own expense.
Orders came hire by the Sidon for the Hecate to proceed home immediately to be paid off. She goes first to Malta, embarks the Duke of Parma, and with him goes to Leghorn, and thence she goes to Portsmouth.

Hermes, Captain Gordon, which was despatched from Simon's-bay some time ago to accompany the Livingstone expedition to the month of the Zambesi River, has returned. She arrived in Algoa bay on the 8th inst., and has brought intelligence from Dr. Livingstone and the members of his exploring expedition. Dr. Livingstone, in a private letter, dated Zambesi River, the 26th of June, 1858, addressed to Sir George Grey, states that the expedition had safely reached the Zambesi, and, having parted company with the Pearl, were about to proceed to Tete in the small steam launch Ma Robert. Some difficulty was experienced in obtaining an entrance for the Pearl into the main stream of the Zambesi. Dr. Livingstone says
"'We first attempted the branch which was described by Lieutenant Hoskins as the most southern and most navigable branch, and though it did not lead us into the Zambesi, we found some 60 or 70 miles of navigable river. ... After searching for some time at the bar of Luabo - which Mr Skead sounded in the Hermes' cutter - we failed to find a passage; but trying by the advice of Captain Gordon, the river Kongone, the bar of which, also, Mr. Skead sounded, we entered, and soon reached the main stream. ... We then let the Pearl go on her voyage to Ceylon, and trust to getting up to Tete by the Ma Robert. We have had no fever yet. Captain Bedingfield has had hard work of it, but he, too, continues well, and we all look forward with interest to meeting with my Makololo, who are still at Tete, though several have died during their stay by smallpox. We shall leave our heavy baggage at Senna.'

"A member of the expedition writing from the Zambesi on the 2d of July, gives the following additional particulars:—
"'The weather has been delightful; no signs of fever; in fact, nothing can be more delusive than the belief that this is the region of death. We found ourselves off the Great Zambesi, in the Pearl, on May 14, but, the river being rough and the wind fresh, we did not attempt to land until the next day, when the Hermes hove in sight: and, as it had been decided by the expeditionists that the great river would be more easily reached by the West Luabo and less risk run than by entering the Zambezi at once, where the bar is shallow and the surf heavy, we decided for West Luabo, accompanied by the Hermes. It was low water when we reached the mouth of the river, with the sea in a state of fury right across its mouth; so we waited till 3 p.m., when, the water having risen six feet, we made a run for it in the Pearl (her captain showing much pluck), and got over the bar (which just broke), 2¼ fathoms being the least water we found. Upon entering the points of the river, a fine sheet of water opened out, the shores of which are densely clad with mangrove and other tropical trees, but the river's banks were quite level, and elevated only two or three feet above the spring tide level. This feature is universal throughout the delta. We anchored for the night, and at day-dawn on Sunday, the 16th, the operation of hoisting out the steam-launch was commenced. I started off with two Kroomen and three of the members of the expedition to survey the estuary, and got astronomical observations, Captain Bedingfield and myself acting as leadsmen. We did our work by 5 p.m., and returned to the Pearl just as the centre and heaviest part of the launch, weighing five or six tons, was going out. All went well, and at sunset we gave three cheers, and joined the fore part of the launch to the middle, and so ended the first day. We found a group of eight hippopotami living in a creek just at our observation spot, and they by no means approved of our intrusion. We fired at them, heard the bullets strike their heads, but they only grunted, sank down and rose again, again to receive another leaden salute with the like indifference. I measured the footprints of these animals on the stiff clayey bank of the river, and found them 15 to 16 inches and 12 inches. Dr. Livingstone declares their flesh to be delicious, and very similar in flavour and delicacy to sucking-pig. I have made arrangements for a hippopotamus ham. Having got all ready for forward work, such as trying the launch, testing the compasses, &c, we left our first anchorage on the 20th of May, with the launch ahead, to lead the way. We soon got aground about seven miles up the river, but did not remain long there, and by 6 p.m. had advanced a good many miles from the sea, where we anchored in six fathoms for the night. We found the river more than anticipation had pictured it to be — broad, deep, and flowing with riverly strength, which raised our hopes far beyond what they had formerly been for success with ease and rapidity. Vain hopes, too soon to be confronted with reality, in the shape of reeds and bulrushes right across the river! The mosquitoes began to be very troublesome. I saw and closely examined six different species, all venomous and brutally ferocious; but we found that by keeping in the middle of the river our sufferings were somewhat alleviated.
"'Very few natives were seen, and they were shy, not daring to approach us, as far as we have yet been.
"'Next day, May 21, we started at daylight, and ran aground at 8. At 10 got afloat and went on, passing a wide branch of the river, several miles from the sea. We continued to proceed, the river getting narrower, until 5 p.m. On the 22d of May we reached a short, sharp turn in the river, and in trying to turn, by putting her bows against the rivers bank, she stuck fast, and at 9 p.m. she was dry as far aft as the foremast, her stem being in four fathoms water and not 20 yards from the other bank. No fever, nor anything that I know of, can be worse than the mosquitoes that night. The spot is called the Mosquito Bend, and the beasts, as if conscious that our onward progress was at an end, feasted with savage prey upon us, curtains, clothing, boots even, proving unavailing against their savage onslaughts.
"'On the 23d of May at day-dawn I left the Pearl in the launch with Captain Bedingfield, to feel for the expected channel to the Zambesi, the fog from the river hardy allowing us to see our way, and the river fast became narrower, and shallow, the middle not exceeding 20 yards. At 8 45 we got aground (the launch floats in two feet), and then we left in a gig to try if a passage at north-west to the Zambesi existed for the launch. The tide being out we did not get far, and as there was not enough water for the gig to turn we backed out and returned to the launch, having floated. At noon we returned for the Pearl, reaching her at 2 p.m.
"' May 24th.— Again left in launch to try for some channel to the Zambesi. We again got aground, when I left in a gig with the geologist and 10 Kroomen, and after pulling, sculling, and poling the boat many miles from the sea we were fairly stopped by reeds and bulrushes in two feet water in every direction. All hope this way was then at an end, and we turned our thoughts to the other branch of the river, from "No Whither Island," and our prow' towards the launch.'

THE LIVINGSTONE EXPEDITION,
(From the Cape Argus of Nov. 20.)

Last week letters were received here from Dr. Livingstone and other members of the exploring expedition to the Zambezi River. The news they contain of the progress of the expedition is alike important and interesting. They had reached Tete, where a supply of coals for the steam-launch was obtained, the first ever taken out of the earth in that country. Livingstone's Makololo friends, whom he had left at Tete previous to going to England, were found to be still there awaiting his return. A letter received by Governor Sir George Grey from the reverend doctor contains some interesting details relative to these matters. Sir George Grey, we may mention, has availed himself of every opportunity to assist the expeditionists, and his kindness is gratefully acknowledged by Dr. Livingstone, who afterwards proceeds to say: —
"I shall commence, without preface, in the middle of things, by saying that I have just come down from Tete to the mouth of the river (Kongone) and will start in a day or two for the same point. The gentlemen there (to Tete), through the influence of Major Lecard, at once acceded to my proposition to get me some coals, and in three or four days we were furnished with a ton and a-half, the very first ever dug in this country. The engineer pronounces them to be of a very good kind, though being from the surface, where, in the bank of the Muatize, they have been exposed to the action of air and floods for ages. Mr. Thornton, the geologist, thinks equally well of them. They contain very little sulphur. I could not ascend the river (Revulene) into which the Muatize flows, to load at the seam itself, as there were only a few inches of water at a bar across its confluence with the Zambezi, but large canoes brought them down. At other times of the year we could have sailed up in this launch, which draws 2ft. 6in., with the greatest ease. There is no end of the finest (specular) iron ore; so with coal (of which many seams or parts of an immense seam crop out) and iron; surely something will yet be done in Africa. This was the first thing of the steamer kind ever seen at Tete, and we were visited with as much interest as is the Leviathan. Foremost among our visitors were my Makololo companions. They grasped my hands and arms convulsively, and lullilooed for joy. About 30 of them have died from small-pox, and six were killed by a rebel chief, who, in defiance of Portuguese authority, holds a stockade at the confluence of the Luenya. This grieves them and me more than any thing. The excuse is he did it in a fit of drunkenness. There were three such rebels, half-caste Portuguese of Goa, who defied the Portuguese. One, who had a stockade at the mouth of the Shire, has just now been conquered by the Governor of Killimane. The war has been against us, though we have gone from one side to the other, without molestation, as friends of both, or rather as English, for it is the English name that was our passport. I came one night to a party after dark, and created an alarm, but that was quelled when I called out 'Mglze.' The river is now nearly at its lowest; and, unlike the muddy rivers of the west, it may be styled one of sand; there is very little mud comparatively. Below Lupata it is spread out to from one mile to three in width, with many islands. In the wide parts I experienced considerable difficulty, and especially in one part called Shigogo; but when we approached Lupata, where all the river is in one body, our difficulties end. At Kelira Basa we shall have another obstacle to surmount. It is described as a number of rocks jutting out of the stream, and narrowing the channel, which is deep and tortuous. There is no waterfall, but we shall go and examine it carefully as soon as we get up; and, this being low water, we shall be able to give a clear idea of the whole. If we could travel as geographers do, with the legs of a pair of compasses, we might have been there long ago. At present we are taking up our luggage from stage to stage, and having been deceived by a false report on the engine of this vessel it is rather slow work. It consumes an enormous quantity of fuel, and half our time, when we have no coals, is spent in woodcutting. This, however, led to our discovering that lignum vitae abounds, and there is also ebony, and teak, or African oak; but we cannot yet say how much. The canoes pass us and look back at the 'Asthmatic,' as I call her now. The vessel herself is all very well, though drawing much more than was predicted; but the engine turns out a wretched piece of gingerbread when worked on wood alone. From the information I formerly received from the Portuguese, I believe that the river could he navigated during only six or nine months in the year, but it is now not far from its lowest, and I begin to think that a vessel drawing only two feet might run the whole year; but this we shall be able to decide next month. It begins to rise again in that following. .. The Portuguese ought to do something in the lighthouse way, and if they would only be at the expense of a few piles at three places all obstructions from shallowness of water would vanish. Taking the river as a whole, there is no lack of water,— witness the reports of Captains Gordon and Bedingfield; and, though it is now eight or nine feet lower, it has still much the same appearance, and always makes me wonder how our easy-chair geographers could imagine it all to come out of a a 'great interior sandy desert.' When it spreads out into several channels a few piles driven in at the part above where the water takes the swing into a shallow one would in one flood effect what is now done by snags,— create a deep channel. A few piles would widen the Kongone canal in one year."

He then suggests that some sort of beacon should be erected at the mouth of the Kongone, to prevent persons being misled and going into the Luabo instead. It appears that when the Pearl first entered the river the steam-launch Ma-Robert was immediately put together although the day was Sunday; and this circumstance has been censured by some persons who were ignorant of the motives which led to it. But the best reply to such criticism is given by Dr. Livingstone himself, who, in the concluding portion of the letter above alluded to, says:—
"When I came in among mangrove swamps ... I pushed on the work with all haste to get out of the Delta. People, I hear, blame me for this; but they would have blamed me much more had I lost nearly all the expedition.'... We take quinine daily, and the short illnesses we have had have partaken more of the character of common colds than fever. Here, in the mangrove swamps, two of our number, who are now better, had, in addition, a dry skin for some hours. But with quinine and care I see no obstacle to passengers going up to Tete from that disease, and there begins the healthy country. Yet it is not to be trusted either; irregularities must be avoided."