From CALIFORNIA our advices are to the 15th of January. The cholera had entirely disappeared. The result of the late State election had been definitely ascertained. In the Senate there is a Whig majority of two, and in the Assembly a Whig majority of nine. This result is deemed important on account of the pending election of U.S. Senator in place of Mr. Frémont. Gov. Burnett has resigned, and Lieut.-Gov. McDougal been installed in his place. Hon. David C. Broderick, formerly of New York, was chosen President of the Senate. Renewed difficulties have occurred with the Indians, and the general impression seemed to be that no friendly arrangement could be made with them. They demand the free use of their old hunting-grounds, and will listen to no proposition which involves their surrender. The settlers, especially on the Trinity and Klamath rivers, suffer grievously from their marauding incursions, and have been compelled to raise and arm companies to repel them. A serious and protracted war is apprehended. The latest arrival brings the report of a discovery of gold exceeding in magnitude any before made. Twenty-seven miles beyond the Trinity River, it is said, is a beach seven miles in extent, bounded by a high bluff. A heavy sea, breaking upon the shore washes away the lighter sand, and that which remains is rich to an unparalleled extent. A company has been formed to proceed to this locality, and the Secretary estimates the sum which each member will secure, at many millions. The whole amount of gold dust shipped at San Francisco during the year 1850, is officially stated at $29,441,583. At least twenty millions are supposed to have gone forward, in addition, in private hands, so that the total product of the mines during the year is estimated at nearly fifty millions. The mines in all quarters continued to yield abundant returns. MEXICO.