JS ew i ork, J uno 1. The following is the report of Licut. Tompkins to Col, Hunter, his superior officer: I have the honor to report that, pursuant to instructions from the Colonel commanding, I left on the evening of the 31st, in cornmand of eompany B cavalry to reconnoitre the country in the viüinity of F i'irfax Courl Uouso. - Approaching the town, the enemy's piuket v;;s surprised and captured. - Severa] documenta were (ound n them, which I endose. On entering the town of Fairfax, my coinrnand was fircd on by rebel troops from windows and house-tops. I charge 1, and drove the rnountod rifles from tlio town. Immediately two or three additional companies carne to their relief, firing on us, which I returned. Perceiving that I tvas largdy outnunbered, I deemed it advisublo to turn, which I did in good order, with five prisoners fully artned andequipped, and two homes. My loss is three men missing, threo 6Üghtly wounded, and gix horses lost. Loss of rebels from twenty to thirty-fivo killed and wounded. From observations I should judge the rebels numberod fully 1,500 men. Gapts. Cary and Fearing, and adjutant Frank, of the New York Fifth, accompanied us, and I regret thatthu former was wounded in the foot. Lieut. G-ordon, Second Dragroons, rendered val uablo service. The prisonors, horses, nrms, and equipments, I have sent to Col. Álandlield.