The report from the Stono to-day gives the usual number of arrivals and departures of vessels. No troops on board any of them.

There is no material change in the fleet at Port Royal. Neither of the missing monitors is reported there.

October 2, 1863.-The fleet inside the bar this morning is composed of the Ironsides, four monitors, two mortar-boats, and twenty-five transports, gunboats, &c.

Moultrie, Simkins, and Haskell have kept up a slow but irregular fire since yesterday morning.

Appearances on Morris Island would appear to indicate permanent occupation by the enemy rather than immediate operations. All the high sand-hills on the southern end of the island are strongly entrenched. Battery Wagner is being extended still more to the southwest, and two siege and two barbette guns are mounted upon one of the faces bearing upon Sumter.

At 8 a.m. the enemy's battery on Gadberry Hill opened upon Sumter and fired 74 shots; 44 struck that work, and two of these penetrated the gorge wall near the old officers quarters. No other material damage to the work, and no casualties reported. A few shots were fired from the same battery at Battery Simkins, which last, together with Moultrie and Battery Cheves, replied slowly to the enemy. A well-directed shell from the latter battery knocked down the enemy's lookout at Wagner.

As a large force of negroes were at work on the bomb-proof at Battery Haskell, it was not deemed advisable to draw the attention of the enemy to that work by firing. Battery Haskell, therefore, continued silent during the day.

October 3, 1863. Since yesterday morning, Batteries Simkins, Cheves, Haskell, and some of the Sullivan's Island batteries have fired 163 shots.

During the past night, the enemy have been busy at Wagner, Gregg, and the sand-hills between those works, where a mortar battery [it is supposed] is rapidly approaching completion. They also appear to be constructing a work on Big Folly Island.

A strange vessel is reported with the fleet this morning, flying the English flag at the main peak and the Federal flag at the fore. It is thought she is a captured blockade-runner. She left for the north in company with the mail steamer at 6 p.m.

The enemy opened this morning from his batteries below Graham's house o Sumter, Johnson, Simkins, and Cheves, directing, however, most of his shots against Sumter. Of the 95 shots fired at that work, 78 struck, but did no material damage, and caused no casualties, as was the case also at the other batteries.

Captain [T. E.] Raysor's company [H], Eleventh South Carolina Volunteers, was relieved from duty in Sumter last night by Captain [E. W.] Lloyd's company [B], Twenty-fifth South Carolina Volunteers.

Thirty-one mortar shells were fired to-day from Simkins and 89 rounds from Cheves, both with reported good effect.

The engineers at Haskell have the bomb-proof at that work under construction.

The following is the result of the mortar firing from that post today, directed against a party of the enemy on Black Island:

First shot, 10 pounds powder, 26,6 seconds, 2 3/8 miles; fell to right, short; did not burst.