FROM THE SAME, May 20th.

The fleet of transports arrived at the falls just above Alexandria, on the twenty-third ultimo, and all passed down to the landing in two or three days. Ten heavy-draft gun-boats were enabled to come down. The Eastport - heavy iron-clad - was aground forty miles above, as was also the Hastings, a sort of independent transport. They were destroyed as effectually as possible. On Tuesday, the twenty-sixth, the whole army had arrived. On or about the first instant, General Smith's command was thrown out several miles, where they remained till the thirteenth, occasionally engaging in slight skirmishes. The remaining infantry and cavalry were camped around the town in every direction. Large parties were engaged in handling quartermaster and commissary stores, in bringing in sugar and cotton and putting it on boats, and in building a dam at the foot of the falls, which was commenced by Admiral Porter, on the thirtieth ultimo. It was a very lively town for two weeks. On the morning of the thirteenth, the last gun-boats were safely over. The dam was an extensive work, and reflects great credit upon the chief engineer, Lieutenant Colonel Bailey, of a Wisconsin regiment.

The entire fleet and army left Alexandria on the thirteenth instant, General Smith's command and some cavalry having the rear.  In the morning, some person, said to be a soldier, set fire to a building on Front street, about opposite to the centre of the town.  Exertions were made by our men to prevent the spreading of the fire. Houses and churches were torn down and blown up, but to no purpose. The fire was raging at 3 o'clock P. M., when the fleet left, and, from what I can learn, fully one-half of the town was consumed.

Of course, there was no apology for the incendiary - nor is there much sympathy in the army for the citizens. They have brought, by their rebellion, an army into their midst, and they must expect not only to submit to the proper and authorized results of military occupation, but also to suffer from unmilitary, unauthorized and wanton acts of those bad men who are to be found in every army. As to making war upon women and children, the only question in my mind as to whether it is most proper to shoot the rebel father and husband, or to burn down the shelter of his family, is one of policy. If, by the latter course, he could be induced for a brief season to forego the exercise of his amiable intention to kill me, and devote his little furlough to the reinstatement of his household goods, I am almost certain that I should incline to the incendiary policy. Besides, who knows but the endearments of conjugal and filial society, might soften the heart of the stern warrior, and bringing his modern Zantippe to insist upon an extension of his furlough, with a slight portion of the vehemence with which she once bade him go out and exterminate the hated ''Yanks," and bring her a nice skull for a drinking cup. These feminine(?) characters are no myths. As to exasperating any body down here, that can't be done. They are just as savage now as they dare be.

The fleet was but little disturbed on its way down. It arrived at the mouth of the Bayou Atchafalaya on the fifteenth, and dropped down to the place of our previous landing, above Simmsport, the next day. A large number of transports were waiting, and a bridge of boats was formed, and the extensive trains of General Banks' commenced crossing at once. Several boats were also engaged in ferrying. On the eighteenth there was sharp fighting between the enemy and the First and Second brigades of Smith's division, with cavalry - the artillery firing said to be heavier than on our lines at Pleasant Hill. Our loss is estimated at fully two hundred - a large portion of the wounds severe. The loss of the Twenty-seventh is four dead and fourteen wounded. Charles Coulon, company H, is among the dead. Hoover, same company, is severely wounded. The enemy was entirely satisfied and did not renew his attacks. We leave this afternoon for Red River landing, where our troops will arrive in the morning, when we take them on board and leave for Vicksburgh. We learn that General Smith is promoted, and we all say deservedly.

H. C. H.