Advices from Texas give accounts of the rapid improvement of the lands on the Brazos River. The troubles with the Indians still continue. A battle between a small company of Texan militia and a band of Indians, took place near the head waters of the Lema River, on the 24th of May: Six Indians were killed, and the remainder driven off. An expedition has been ordered by Gen. Harney, to aid the Indian agents in their demand for the release of white prisoners in captivity. A train, composed of 170 wagons, with a large escort, left San Antonio for El Paso on the 7th of May. A company of Americans, while crossing the Rio Grande to attend a ball at Rima, were fired upon by a party of Mexican soldiers. Two of the American soldiers were severely wounded, and the Mexicans apologized for the act on the ground of its being a mistake.

News from Santa Fe to the 1st of May has arrived. On the 2d of April, Governor Calhoun consummated a treaty with Francisco Chacon, principal chief of the Apaches east of the Rio Grande. The savages agreed to give up what stolen property had been in their possession for the previous eight months, and to settle in towns, provided teachers and implements of husbandry were furnished them. As might have been expected, this treaty was broken within three weeks of its adoption, although Chacon bound himself to maintain the peace, on penalty of forfeiting his head. Fifteen companies of the U.S. troops were to leave Santa Fe on the 10th of May, upon a campaign against the Navajo Indians. This movement was considered necessary, on account of the serious injury which the health of the soldiers had sustained from the inactivity of their mode of life.

Governor Calhoun issued a proclamation on the 23d of April, appointing the 19th of May for the election of members of the Legislative Assembly. The first session of the Legislature was to commence at Santa Fe on the 2d of June. The Mexicans were well pleased with the new Government, since it removed the power from the hands of the military. Business was very brisk at Santa Fe, and a number of mills were in the course of erection in the neighborhood. The census of the territory, taken by direction of the Governor, shows a population of 56,984, in addition to the Indians. The Boundary Commissioners were on the Rio Grande, near Dona Ana, and had decided to place the corner-stone six or seven miles below that place.

The news of the formation of a Territorial Government for Utah, and the appointment of Brigham Young as Governor, was first received at the Salt Lake, by way of California. The General Assembly of the Church for the State of Deseret, have transferred all their powers to the Territorial Government, and adjourned. The "Quorum of Seventies" had agreed to erect an extensive rotunda in the Salt Lake City, to be called the "Seventies' Hall of Science." The Mormons have established a colony in Iron County, about 250 miles nearly south of the Salt Lake City. Several families, with 130 men and supplies of all kinds, under charge of Elder Geo. A. Smith, left on the 7th of December, and when last heard from, they had 1600 acres cleared, and 400 sown with grain. Elders Lyman and Rich left early in March with 150 wagons, to form another settlement on the Colorado, on the Californian line. The Mormons design establishing a continuous line of stations on the Pacific on this route.

The steamers which left San Francisco on the 15th of April and the 1st of May, carried away $3,000,000 in gold dust, nearly all of which was shipped to the Atlantic States. The news from all parts of the gold region is unusually favorable. The rains which came on towards the end of March continued for two weeks, and furnished an abundant supply of water for the dry diggings. The piles of earth which had been heaped up during the winter, were yielding excellent returns. In the higher ranges of the mountains there had been heavy falls of snow, which had cut off the supplies of some of the remote diggings, and several persons were frozen to death near the head waters of Feather River. The rich placers discovered in this region have attracted many thousands of miners; and the trail through the snows was lined with the carcases of mules which had perished from the cold. On account of the scarcity of supplies, board had risen to $56 per week.

Important discoveries have been made in Shaste Valley, in the northern part of the State. One thousand acres were tested, and found to yield ten cents to the panful of earth. The first discoverers averaged $80 apiece daily. The diggings differ from all others in the circumstance of all the earth containing gold down to the bottom rock, which is struck at a depth of four feet. The gold is found in coarse grams, interspersed with large lumps. An extensive emigration had already set towards the new placer. The Volcano diggings continue to give large returns; while there is no diminution in the yield of the old localities on the American Fork, the Stanislaus and the Mariposa. The quartz veins on the latter river and in the neighborhood of Nevada City, give proof of astonishing richness; but the gold is generally found in such fine particles, that not more than half of it can be collected by any machinery which has yet been brought into use. Veins of silver ore, which promise to be very rich, have been discovered on Carson's Creek.

The Californian Legislature adjourned on the last day of April, after a session of four months. Among its last acts was the passage of a law, exempting homesteads and other property from forced sale in certain cases. It also passed a Usury Law, fixing interest at ten per cent., and allowing eighteen per cent. by special agreement. Party politics have attained a height scarcely known in the older States at present. The City election in San Francisco was very hotly contested, but finally resulted in the choice of all the Whig candidates, except two. Both parties are marshalling their forces for the coming State election. The prominent candidates for Governor, are Major Roman, the present State Treasurer, with the Democrats, and Major Pearson B. Redding with the Whigs.

A body of Indians have been committing depredations on the Salinas Plains, near Monterey. They have killed three persons near the town of San Luis Obispo, robbed all the ranches, and driven away the horses from San Antonio to San Miguel. According to the treaty made with the Nevada Indians by the U. S. Commissioners, six tribes, numbering in all 1500 persons, have been removed to a tract of land twelve miles square, between the Merced and Tuolomne Rivers, which is secured to them for ever. In the vicinity of Los Angeles, the tribes still continue their depredations. Lynch law still remains in force in all the mining districts. A band of five Mexicans, who had been detected stealing cattle on the San Joaquin River, were tried in a summary manner, and all executed.

A project has been started to supply San Francisco with water from a lake called "Mountain Lake," a few miles from the city. It is described as a body of pure water, a mile in circumference, and 153 feet above the sea. A line seventy-five feet long, was dropped into the centre without finding bottom. It is estimated to furnish twenty-five millions of gallons of pure water daily. In the neighborhood of San Francisco, San José, Sacramento City, Sonoma and Bodega, large tracts of land have been brought under cultivation: and the harvest of grain and vegetables will this year go far towards supplying the wants of California. Nearly all kinds of vegetables attain a size and flavor which are not equalled in any other part of the world.

It is rumored that an expedition is about being raised in the southern part of California, for the purpose of invading the Mexican province of Lower California. A certain Gen. Morehead is said to have left with a force of two hundred men, well armed and provisioned. There is also talk of similar movement, having reference to the State of Sonora.