As I understand an expedition is about to go down the river, for the purpose of reducing Vicksburg, I have prepared the enclosed map in order to demonstrate more clearly the obstacles to be encountered in the contemplated assault. In the first place, it is impossible to take Vicksburg in the front without too great a loss of life and material, for the reason that the river is only about half a mile wide, and our forces would be in point-blank range of their guns, not only from their water-batteries which line the shore, but from the batteries that crown the hills, while the enemy would be protected from the range of our fire. By examining the map I enclose, you will at once perceive why a place of so little apparent strength has been enabled to resist the combined fleets of the Upper and Lower Mississippi. The most economical plan for the reduction of Vicksburg now, is to push a column from Memphis or Corinth down the Mississippi Central Railroad to Jackson, the capital of the State of Mississippi. The occupation of Jackson, and the command of the railroad to New Orleans, would compel the immediate evacuation of Vicksburg, as well as the retreat of the entire rebel army east of that line; and by another movement of our army from Jackson, Miss., or from Corinth to Meridan, in the State of Mississippi, on the Ohio and Mobile Railroad, especially if aided by a movement of our gun-boats on Mobile, the Confederate forces, with all the disloyal men and slaves, would be compelled to fly east of the Tombigbee. Mobile being then in our possession, with 100,000 men at Meridan, would redeem the entire country from Memphis to the Tombigbee River. Of course I would have the gun-boats with a small force at Vicksburg, as auxiliary to this movement. With regard to the canal, Vicksburg can be rendered useless to the Confederate army upon the very first rise of the river; but I do not advise this, because Vicksburg belongs to the United States, and we desire to hold and fortify it, for the Mississippi River at Vicksburg and the Vicksburg and Jackson Railroad will become necessary as a base for our future operations. Vicksburg might have been reduced eight months ago, as I advised after the fall of Fort Henry, and with much more ease than it can be done to-day. It will be recollected that after a month's attack upon Vicksburg, commencing June 28, 1862, by the combined Farragut fleet, Porter mortar flotilla and the gun-boat fleet under Capt. C. H. Davis, the bombardment of the city was suspended, it being found impossible to capture and hold it with the forces at command.