THE DEMONSTRATION ON TUESDAY NIGHT. – The Union men of this city have shown themselves to be as superior to their opponents in numerical force as they are in all the principles of patriotism and human rights. According to the Wide Awake procession early this month, and the Union display of Tuesday evening, it is evident that, in the coming election, this city will poll an overwhelming majority against the rail splitter. The majority will be in the proportion of 10,000 to 40,000 – the respective numbers of the two street demonstrations. The procession on Tuesday night was one of the most brilliant ever seen in New York. Considering the unfavorable circumstances under which it was organized and carried into perfection, it must be universally admitted that it was the most enthusiastic outburst ever offered to the Union cause. It is now the duty of the conservative merchants of the city to see that all the bills arising out of this popular manifestation should be discharged, and that the burden shall not be thrown upon the shoulders of Captain Rynders. The demonstration has conclusively shown that New York is truly a national city.