Having accomplished the task assigned us, we in return feel a degree of anxiety as to the characters who will probably be candidates for those two important offices. I believe it is pretty generally understood that Mr. Jefferson is contemplated for President. But who is to fill the Vice-President's chair? I should be highly gratified in hearing your opinion on this subject; if secrecy is necessary, you may rely on it; and, sir, as I have no personal views, you will readily excuse my stating the present apparent wishes and feelings of the Republican party in this city. It is generally expected that the Vice-President will be selected from the State of New York. Three characters only can be contemplated, viz., Geo. Clinton, Chancellor Livingston, and Colonel Burr. The first seems averse to public life, and is desirous of retiring from all its cares and toils. It was therefore with great difficulty he was persuaded to stand as candidate for the State Legislature. A personal interview at some future period will make you better acquainted with this transaction. In addition to this, Mr. Clinton grows old and infirm. To Mr. Livingston there are objections more weighty. The family attachment and connection; the prejudices which exist not only in this State, but throughout the United States, against the name; but, above all, the doubts which are entertained of his firmness and decision in trying periods. You are well acquainted with certain circumstances that occurred on the important question of carrying the British treaty into effect. On that occasion Mr. L. exhibited a timidity that never can be forgotten. Indeed, it had its effect when he was a candidate for governor, though it was not generally known. Colonel Burr is therefore the most eligible character, and on him the eyes of our friends in this State are fixed as if by sympathy for that office. Whether he would consent to stand I am totally ignorant, and indeed I pretend not to judge of the policy farther than it respects this State. If he is elected to the office of V. P., it would awaken so much of the zeal and pride of our friends in this State as to secure us a Republican governor at the next election (April, 1801). If he is not nominated, many of us will experience much chagrin and disappointment. If, sir, you do not consider it improper, please inform me by post the probable arrangement on this subject. I feel very anxious. Any information you may wish relative to our election I will at all times cheerfully communicate. With sentiments of respect, &c. GALLATIN TO HIS WIFE. PHILADELPHIA, 6th May, 1800. --The New York election has engrossed the whole attention of all of us, meaning by us Congress and the whole city. Exultation on our side is high; the other party are in low spirits. Senate could not do any business on Saturday morning when the intelligence was received, and adjourned before twelve. As to the probabilities of election, they stand as followeth: