NEAR BETHESDA CHURCH, June 1, 1864-8 p.m. (Received 7 p.m., 2nd.)

At about 5 o'clock we heard the cannon of Sheridan, and soon after Wright and Smith attacked Lee's right wing with their whole force. They moved from Cold Harbor in the direction of Mechanicsville. Judging from the sounds of artillery and musketry, the fight was furious. Our latest report from Wright is dated 7.30. He says he has great advantages and has taken many prisoners. At about 6 o'clock Warren attacked in the center, but apparently not with much force. His firing is that of a lively skirmish. Immediately upon Wright's attack the enemy moved out on his left against Hancock, as if to try what strength we had in that direction. He was decisively repulsed. Hancock followed up the repulse, but was not able to get over the rebel works, and fell back to his own lines. No detailed report of any of these operations has yet been received. Ames' division, of Smith's corps, remains at the White House, waiting to be relieved by Abercrombie.

C. A. DANA.

Honorable E. M. STANTON,

Secretary of War.

NEAR BETHESDA CHURCH, June 2, 1864-6 a.m. (Received 8.45 p.m.)

The details of yesterday's operations, though as yet but imperfectly reported, are more favorable to us than my dispatch of last evening indicated. It appears that the rebels three times assaulted the lines of Griffin, and they came up in three lines. They were terribly slaughtered by canister, and went back in disorder every time. In Wright's attack, Ricketts, whose division held the right of the corps, took more than 500 prisoners, mainly from Beauregard's corps, carried the rebel works before them, but withdrew afterward, on account of an enfilading infantry fire. It appears that Sheridan did not attack, his order not having reached him in time, and his troops being scattered. He will go in this morning. What we supposed to be his guns were those of Wright's left division. Hancock moved during the night to Cold Harbor, where his advance arrived about daylight. His rear is now marching past these headquarters. In conjunction with Wright and Smith, he will this morning fall upon Lee's right. During the night our forces in that quarter were repeatedly assailed by strong rebel columns. An attack at 10 p.m. is described by a staff officer of Smith's as more determined and valiant than any fighting during the day. The rebels gained nothing by any of these efforts. Warren and Burnside are ordered to open as soon as they hear that the three corps on our left have begun the battle. General Lockwood, commanding Warren's left division, when ordered to attack yesterday afternoon, though in position near the enemy, and with the sound of the fight on his right and left, lost his way, and was finally discovered with his division 2 miles in the rear. He has been relieved, and Grant will recommend his dismissal from the service. General Wilson reports at 6.30 p.m. yesterday that he had destroyed the South Anna bridges of both railroads, but one of his brigade was then wholly engaged with an infantry force sent from here by Lee yesterday