Washington, May 30. To Ma.t. Gen. Dix : A dispatoh frorn General Grant bas just been reoesved, dated yesterday, 29t!i, at Hañovertown, and states that tho army has sncce.-'sfuüy crossed over the Pamunky", and now oeeupies a fiont nbout three railes soulh of the river. Yesterday tvvo divisions of' our cavalry had an epgagement with tho enemy south of Hawes' store, driviug hirn about a mile, upon what appoars to bu his new line. We will fitid out all about it to day. Our loss ia tho cavalry engngement was 350 killed and wounded, oí' whoni but lorty-fonr are ascertuined to have been külnd, we having driven the enemy. Most of their killed, missing aud wounded fell rato our handa. Another o-fficial dispatuh, dated yesterday 2 o'clock, P. M. details the movernenls of the several corps then in progress, but up to that time there was no engiigcment, Bfirlief dispatches from heudquarters had been sent, but failed to reach WashingtoD.