EAST CUMBERLAND AGRICULTURAL SOCIETY. -- It will be seen, on referring to the advertisement of the Society's show at Carlisle, which will be found elsewhere that Messrs. DAWSON and BUSHBY offer a premium for turnips grown by guano.  This new manure has, during the present season, attracted very great attention; and, should all the promises held out be realized, it will become of vast importance.  In order that as correct information as possible may be had, it is desirable that the judges should ascertain, and report all particulars that may be interesting, such as the quantity used per acre, the mode of applying it -- its relative merit as compared with other manure, and in the same field, or on the same farm, notice being taken of the composition and quality of the soil, &c.  It will be observed that the entry for potatoes and turnips closes on the 2nd, and for stock, on the 9th of September.