The course of events which disabled Mr. Polk as a candidate proved equally decisive against all the members of his cabinet; and by the process of exclusion rather than by an enthusiastic desire among the people, and still less among the leaders, General Cass was selected by the Democratic Convention as candidate for the Presidency, and William O. Butler of Kentucky for the Vice-Presidency. The Democracy of New York, in consequence of the divisions arising under the governorship of Mr. Wright, sent two full delegations to the convention, bearing credentials from separate organizations. The friends of Mr. Marcy bore the name of Hunkers; the followers of Mr. Wright ranged themselves under the title of Barnburners,-- distinctions which had prevailed for some years in New York. It was in fact the old division on the annexation of Texas, and now represented the pro-slavery and the anti-slavery wing of the Democratic party. The National Convention sought in vain to bridge the difficulty by admitting both delegations, giving to them united the right to cast the vote of the State. But the Barnburners declined thus to compromise a principle. On a question of bread, the half-loaf is preferable to starvation, but when political honor and deep personal feeling are involved, so material an adjustment is not practicable. The Barnburners retired from the convention, disclaimed all responsibility for its conclusions, and proceeded in due time to organize against the ticket of Cass and Butler. The Hunkers, left in the convention as the sole representatives of the New-York Democracy, were startled at the situation and declined to vote. They were anxious that the nomination of Cass should not appear to be forced on the Barnburners by the rival faction. It thus happened that New York, which for twenty years under the skillful leadership of Mr. Van Buren had dictated the course of the Democracy, was now so shorn of influence through the factions engendered by his defeat, that a Presidential nomination was made, not only without her lead, but without her aid or participation. CASS BOLTED BY VAN BUREN'S FRIENDS.