_September 13._--An advance of the Union army was ordered yesterday by its Chief, in which the cavalry was to take a prominent part. Orders were issued accordingly last evening, and every needed preparation made for our work. At an early hour this morning the entire cavalry corps was on the march. In order that the enemy might not be prematurely warned of our design, the several commands were ordered to make as little noise as possible. Consequently the bugle-calls were dispensed with, and commanders made use of their voices, and in some instances the orders were conveyed from rank to rank in a whisper. The three great divisions of the corps were to cross the river as follows: Gregg's, at Sulphur Springs; Buford's, at Rappahannock Bridge; and Kilpatrick's, at Kelly's Ford. BRANDY STATION NO. 3. At six o'clock the Harris Light plunged into the river at Kelly's Ford, leading the advance. A strong detachment of Stuart's cavalry, consisting of pickets and reserves, opposed our crossing with dogged pertinacity, but finally, yielding to our superior numbers and to the deadly accuracy of our carbines, gave way. He then advanced in the direction of Brandy Station. The farther we advanced the stronger grew the ever-accumulating force of the enemy, who disputed every inch of ground with great stubbornness. On arriving near the Station we found the enemy in strong force, with artillery posted on the surrounding hills. We saw clearly that a third cavalry fight was destined to be fought on this historic field, and we began to make preparations for the onset. It was my fortune to lead the advance company in the first charge. Three men and four horses were killed and wounded in this company by the first discharge of the enemy's artillery, whose fire was terribly accurate. But we had not been fighting long before the other divisions joined us. At their approach great enthusiasm among our boys prevailed. Before our combined force the enemy was swept from those plains like chaff before the whirlwind. They fled in the direction of Culpepper, a naturally strong and now fortified position, where we knew we must soon encounter the Rebel chivalry _en masse_ upon their chosen field. FIGHT AT CULPEPPER COURT HOUSE.