Royal.—­On May 12th I made my first proposal of the Reflex Zenith Tube.  The principle of it is as follows:  Let the micrometer be placed close to the object-glass, the frame of the micrometer being firmly connected with the object-glass cell, and a reflecting eye-piece being used with no material tube passing over the object-glass:  and let a basin of quicksilver be placed below the object-glass, but in no mechanical connection with it, at a distance equal to half the focal length of the object-glass.  Such an instrument would at least be free from all uncertainties of twist of plumb-line, viscosity of water, attachment of upper plumb-line microscope, attachment of lower plumb-line microscope, and the observations connected with them:  and might be expected, as a result of this extreme simplicity, to give accurate results.—­A considerable error was discovered in the graduation of Troughton’s Circle, amounting in one part to six seconds, which is referred to as follows:  ’This instance has strongly confirmed me in an opinion which I have long held—­that no independent division is comparable in general accuracy to engine-division,—­where the fundamental divisions of the engine have been made by Troughton’s method, and where in any case the determination by the astronomer of errors of a few divisions will suffice, in consequence of the uniformity of law of error, to give the errors of the intermediate divisions.’—­The method of observing with the Altazimuth is carefully described, and the effect of it, in increasing the number of observations of the Moon, is thus given for the thirteen lunations between 1847, May 15, and 1848, May 30.  ’Number of days of complete observations with the Meridional Instruments, 111; number of days of complete observations with Altitude and Azimuth Instrument, 203.  The results of the observations appear very good; perhaps a little, and but a little, inferior to those of the Meridional Instruments.  I consider that the object for which this instrument was erected is successfully attained.’—­Being satisfied with the general efficiency of the system arranged by Mr Brooke for our photographic records (of magnetical observations) I wrote to the Admiralty in his favour, and on Aug. 25th the Admiralty ordered the payment of L500 to him.  A Committee of the Royal Society also recommended a reward of L250 to Mr Ronalds, which I believe was paid to him.—­On May 1st the last revise of the Lunar Reductions was passed, and on May 5th, 500 copies were sent for binding.—­In this year Schumacher and I refused a medal to Miss Mitchell for a Comet discovered, because the rules of correspondence had not been strictly followed:  the King of Denmark gave one by special favour.—­In this year occurred the discovery of Saturn’s 8th Satellite by Mr Lassell:  upon which I have various correspondence.—­On the 18th of December the degree of LL.D. was conferred upon me by the University of Edinburgh.—­The Ipswich Lectures:  A wish had been expressed that