Mr. Pichon with his respects to Mr. Madison forwards a note1 which after mature deliberation he has thought it his duty to address on the subject of the notes of His C. M. minister to the U. S. in relation to Louisiana. Mr. Pichon expects that it will appear but proper to the President of the United States that Mr. Pichon’s vindication of his Government should accompany, wherever they may appear, the notes which, in Mr. P.’s opinion convey an attack on the faith and dignity of the Same Government.