On the 5th of August Mr. PEARCE, of Md., introduced a bill, making proposals to Texas for the settlement of her western and northern boundaries. It proposes that the boundary on the north shall commence at the point where the meridian of 100° west longitude intersects the parallel of 36° 30' north latitude, and shall run due west to the meridian of 103° west longitude: thence it shall run due south to the 32d degree north latitude, thence on the said parallel to the Rio del Norte, and thence with the channel of said river to the Gulf of Mexico. For relinquishing all claims to the United States government for territory beyond the line thus defined, the bill proposes to pay Texas ten millions of dollars. The bill was debated for several successive days, and on the 9th was ordered to be engrossed, yeas 27, nays 24, and received its final passage on the same day, yeas 30, nays 20, as follows: YEAS.--Messrs. Badger, Bell, Berrien, Bradbury, Bright, Cass, Clarke, Clemens, Cooper, Davis, of Massachusetts, Dawson, Dickinson, Dodge, of Iowa, Douglas, Felch, Foote, Greene, Houston, King, Norris, Pearce, Phelps, Rusk, Shields, Smith, Spruance, Sturgeon, Wales, Whitcomb, and Winthrop--30. NAYS.--Messrs. Atchison, Baldwin, Barnwell, Benton, Butler, Chase, Davis, of Mississippi, Dodge, of Wisconsin, Ewing, Hale, Hunter, Mason, Morton, Seward, Soulé, Turney, Underwood, Upham, Walker, and Yulee--20. Thus was completed the action of the Senate on the second of the great questions which have enlisted so much of public attention during the past few months.--On the 14th the bill providing a territorial government for New Mexico was taken up. Mr. CHASE moved to amend it by inserting a clause prohibiting the existence of slavery within its limits, which was rejected, ayes 20, nays 25. The bill was then ordered to be engrossed for a third reading, which it had, and was finally passed.