The mail brought letters and newspapers, all in so hopeful a tone and with so much favoring private intelligence that I could not resist a feeling of elation for the day. The appearances at Vicksburgh are certainly much better than they ever were before. Yet we have been so often disappointed that it seems as if we never could have a decisive success and it was idle to expect one. The case of the Alexandra comes up in the Exchequer Court today. Mr Evarts came in at luncheon and reported progress. Mr Dudley came in the evening and did the same. It did not strike me very favorable. The Judge is quite old and the case is of a novel character. The Attorney General is by no means a very strong person. So that on the whole we are in the hands of the Philistines. Mrs Adams had her second reception this evening. After more than half the number of last Monday. The experiment is always doubtful in London from the small circle out of which we draw.