Dr. Brinkley, of the Observatory of Dublin, having noticed for several years past a periodical deviation of several fixed stars from their mean places, strongly indicating the existence in them of annual parallax, the author was induced to institute a series of observations upon the subject, the results of which are submitted to the Royal Society in the present communication. Being unable to devote the mural circle, erected at the Royal Observatory in 1812, entirely to this investigation, the Astronomer Royal employed two ten-feet telescopes, fixed to stone piers, and directed to the particular stars whose parallax was suspected, and furnished with micrometers for the purpose of comparing them with other stars passing through the same field. The question of parallax is, theoretically speaking, rather curious than important; but with regard to the state of practical astronomy the case is very different, and, as far as relates to the natural history of the sidereal system, it is a subject of interest to ascertain whether the distances of the nearest fixed stars can be numerically expressed from satisfactory data, or whether it be so immeasurably great as to exceed all human powers either to conceive or determine. The principal stars observed by Dr. Brinkley were, α Lyræ, α Aquilæ, α Cygni.