On the 2d day of July, the army crossed the Potomac at Williamsport, by means of the ford. The crossing was commenced at daylight, and consumed the whole of the day. Just before daylight, a little passage at arms occurred on the Virginia side of the stream, the companies who had been thrown over the night before as pickets having been fired on by a detachment of the ‘Berkeley Border Guard,’ and returning the fire promptly. But this served only to stimulate the already keen energies of the Federal forces, who waded knee-deep through the clear Potomac, and trudged along over the ‘sacred soil’ with a willingness unchecked by the cold nor’wester that raged on that July morning. That portion of Berkeley County, Virginia, which lies opposite to Willlamsport, is called ‘the Neck,’ being in the shape of a horse-shoe, and nearly surrounded by the detour of the Potomac. The turnpike leading from Williamsport to Martinsburg and Winchester traverses the whole length of ‘the Neck;’ and it was on this road that the advance guard of the division, Abercrombie’s Brigade, took its line of march, a brush with the rebels being momentarily expected. The first view of their pickets, after leaving Williamsport, was obtained at Falling Waters, by which sonorous appellation the Virginians designate a small and pretty mill-pond, which loses itself over the dam of a solitary grist-mill, within a stone’s throw of the Potomac. Here was a strong natural position, and an excellent place for waging a defensive war, if the rebels had been so disposed. But they did not make a stand till a point was reached a mile south from Falling Waters, and about five miles from Williamsport, where their skirmishers opened fire at 9.15, A.M. The skirmish which ensued, and which has since been styled the Battle of Falling Waters, was sustained on the part of the Federals by Abercrombie’s Brigade, consisting of the 1st Wisconsin and the 11th Pennsylvania regiments, McMullen’s Philadelphia company of Independent Rangers, the Philadelphia City Troop of cavalry, and Perkins’ Field Battery of six guns. This force speedily dislodged a superior force of the enemy, and pursued them for two miles, as far as the hamlet of Hainesville, where orders from Gen. Patterson to cease the pursuit allowed the rear-guard of the rebels to elude their grasp. The contest and the chase lasted but two hours, and at noon the advance guard encamped at Hainesville. The remainder of the