The following articles are transcribed from the New York Times, dated June 30, 1863:

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MEMPHIS, Friday, June 26,

CAIRO, Monday, June 29.

 

Advices from GRANT'S army to the noon of the 22d, say the siege was progressing vigorously. On the 20th the cannonading was kept up from morning until noon by the army and navy, but elicited no reply from the garrison. A large number of buildings were destroyed; bat beyond that it is not ascertained that much damage was done. Deserters report that the Tennessee and Georgia troops are much dissatisfied, and talk of refusing to fight any longer. JOHNSTON's movements are still wrapped in mystery. He was last reported to be moving south. His forces are all witdrawn beyond the Big Black. Gen. TAYLOR is said to have hung all the Federal prisoners taken at Milliken's Bend. Retailtory measures will probably follow. Very heavy rains in this vicinity have raised all the streams to overflowing, and very few movements on either side have been made. JOHN PARK, Conservative, was elected Mayor of Memphis yesterday.

 

WALNUT HILLS, NEAR VLCKSRURGH, Tuesday, June 23.

 

The rebels in front of Gen. SHERMAN, on our left, came out of their works on Monday night, and temporarily drove our pickets from their rifle-pits; but they were driven back by our batteries. It is thought that they are feeling their way to cut out, or to send out messengers in the melee. There has been considerable firing along the lines to-day. Siege guns are arriving in large quantities. The abulances sent out for our wounded toward Jackson, on Sunday, have returned to the Big Black Bridge, having been turned back at Edward's station. No rebel camps are nearer than that point.

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The London Times, in an editorial on the position of affairs at Vicksburgh, as advised by the China, argues that the first accounts had not been followed by the successful results expected, and with every day GRANT's position was becoming more critical. The article says: "The fortunes of the Federal arms at the last date were balancing between a temporary check and total failure. All the details leave an impression that unusual energy had secured very slight results." Later advices from Vicksburgh were ??? awaited in England. The more especially as it was reported through private channels, and extensively credited, that Vicksburgh had been relieved, and that GRANT's army had been surrounded by the Confederates. AN ADDRESS FOR SYMPATHY. The London morning Herald publishes a lengthy appeal from nearly one hundred ministers, of all denominations, in the Confederate States, seeking to enlist English sympathy in the Confederate cause. It is stated that the address originated from no political source whatever, but from a conference of ministers held at Richmond. The address takes the ground that the restoration of the Union is impossible.