Hon. H. W. HILLIARD, Member of Congress from Alabama, has written a letter declining a re-nomination, and discussing at some length the present condition of public affairs. The events of the past year, instead of impairing the strength of our political system, have, in his judgment, really served to demonstrate it. There is a growing conviction in the mind of the whole nation that the Constitution must be adhered to in its pristine spirit, and that, while it is adhered to, the republic will endure. He had no fear that the extension of our limits would enfeeble us. Our progress is the spread of a great family, all bearing with them the law, the traditions, the sympathies, and the religion of those from whom they sprung. The true way of perpetuating our Union is by multiplying the means of intercommunication, by making taxes as light as possible, by reducing postage, multiplying railroads, and bringing the Pacific coast nearer to us by the early construction of one of those great highways. The scheme of retaliation, lately projected, of discriminating against the products of other States must be abandoned, and our whole legislation, State and National, must be guided by a comprehensive, national, and patriotic spirit. "These States must regard each other as kindred States; the Constitution must be recognized in all of them as the supreme law; and the acts of Congress, passed in accordance with its provisions, must be obeyed, and we must fix in our minds and in our hearts the idea that, as we have had a common origin, we must have a common destiny." The measures of the last session must be regarded as a final adjustment of the disturbing questions growing out of slavery. Mr. H. exhorts to a conciliatory and a patriotic spirit. "Let us forbear," he says, "any hostile acts on our own part. I certainly desire to see in the midst of the great agricultural regions of the South a varied industry, which shall rival that of the North, and which shall spread over our fertile plains all the embellishments which wealth and a high civilization can bestow. I desire, too, to see a direct trade with foreign countries carried on through Southern ports. But I desire to see all this brought about by the enterprise and the energy of our people, entering into a bold and generous competition with those of the other States. We should seek to make Alabama a great and wealthy State; and we can do this by the vigorous development of our resources. Our fertile soil, our noble streams, our great cotton crop, our exhaustless mineral wealth, our population intelligent, industrious, enterprising, and religious--these will enable us to advance with a steady and rapid march in civilization, without resorting to legislative expedients to tax the products of other States associated with us in a common Government, one of the great objects of which is, to keep open the channels of intercommunication."