Kentucky."----A large public meeting was held in New York, on the 5th of March, of those in favor of the nomination of Mr. Webster for the Presidency, subject to the decision of a National Whig Convention. Mr. George Griswold presided. An address was adopted rehearsing the public history of Mr. Webster, and referring to his services to the country in the various public offices which he has held.----A Whig State Convention in Indiana adopted resolutions nominating General Scott for the Presidency.----Washington's birth-day was celebrated at the National Capital by a banquet, got up mainly by members of Congress. Senator Stockton presided, and speeches were made by several gentlemen--mainly directed against the policy of intermeddling to any degree or for any purpose in the affairs of foreign nations. Mr Clay, whose illness prevented his attendance, wrote a letter, saying that the serious efforts made to subvert the policy of neutrality established by Washington, called for energetic measures of resistance. The attempts made to induce this country to plunge, by perilous proceedings and insensible degrees, in the wars of Europe, rendered it proper to recall attention to his principles by celebrating his birth-day. From CALIFORNIA we have intelligence to the 2d of February. Col. JOHN B. WELLER (Democrat) has been elected United States Senator in place of Col. Frémont. He was once candidate for Governor of Ohio and more recently chief of the Mexican Boundary Commission.----Governor Bigler has sent to the Legislature a special message, concerning the financial affairs of the State, in which he urges upon the Legislature the early adoption of measures to relieve the burden of the State's liabilities, and exhibits the amount of her indebtedness. According to the Controller's report, $1,000,000 still stands against the State from the expenses of last year's military expeditions. The aggregate indebtedness, civil and military, of the State, on the 31st December was $2,242,339.74.----There had been no further disturbances from the Indians, though further precautions against them had been taken by sending troops into their neighborhood.----Hon. T. B. King has published a letter recommending the relinquishment of the public lands to actual settlers, and the confirmation by Congress of the rules established by the miners themselves, defining the rights of those who may be employed in the collection of gold, or who may invest capital in machinery for the purpose of working the vein mines.----Intelligence from the mining districts continues to be encouraging. The quartz mining companies are generally doing well, though from defects in machinery some failures have occurred. New discoveries continue to be made.