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The Battle at Bull Run.

WASHINGTON, July 10 — The first division of the grand army, under [unknown] General Tyler, advanced yesterday afternoon from Centreville to a place called Bull Run, about three miles further west. The object of this movement was to get a force around Manassas Junction, upon the Manassas Gap Railroad, as Buckland, so as to turn Gen. Beauregard's flank. On approaching the Run, a fire was opened upon our column from a masked battery on the opposite side of the stream. Capt. Ayres, with his battery of light artillery, regulars, was at once ordered up and obeyed with the most remarkable promptitude. He placed his battery in position with incredible quickness, and replied to the rebels by an incessant firing for thirty minutes. The rebels were then reinforced by five additional regiments, supposed to have been sent up from Manassas Junction. Gen. Tyler immediately dispatched a messenger to Gen. McDowell, at Centreville, for reinforcements. The rebels on being reinforced, charged upon our lines with fixed bayonets. Their charge was most gallantly received by the New York sixty-ninth and seventy-first regiments, who drove them back with severe loss. Fifteen on our side were killed, and fifty-three severely wounded. A number more were slightly wounded.

Deserters who have arrived in the federal camp, say that the rebel loss was twenty-two killed and ninety wounded.

All the available troops in Washington have been sent across the Potomac to-day. Reports were received here this morning of severe fighting along our advanced lines, which have formed the basis for the rapid transmission of these troops. General Scott has no fears of the result of any engagement our troops may have. The troops now engaged have advanced without tents or baggage. Provisions were sent to them today, by railroad, to Berks station.

Gen. McDowell has sixty cannon and fifteen hundred cavalry in his column. A full battery of light artillery has just crossed the Long bridge, from this city with the horses on a gallop. Their particular destination is unknown.

The war department is now in constant communication with the army by telegraph at Centreville. Every movement is reported here immediately. The rumors that have been circulated this afternoon that our troops have met with reverses, and that the secretary of war had announced that the news received from the army was unfavorable, are all fiction.