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 Lausanne, then at Stuttgart, devoted his entire energies to preparing for publication the copious materials at his disposal. His sole recreation was an hour's music with his family in the evenings ; for he played the violin well, and was as an ardent admirer of Beethoven. His insufficient private resources were meantime supplemented by a small pension from the Rajah of Travancore, in whose service he had been a loser in point of interest upon sums advanced for scientific purposes. In 1873 he came to live in London, where in the year following he issued a quarto volume entitled 'Observations of Magnetic Declination made at Trevandrum and Agustia Malley in the Observatories of his Highness the Maharajah of Travancore in the years 1852 to 1869.' It contains an exhaustive and highly valuable discussion of the various modes of solar and lunar action on magnetic declination, of which element alone upwards of 300,000 reduced observations were available from the thirteen years of his administration. The publication, however, went no further, and Broun had the mortification of seeing his life's work left incomplete, and the fruits of his anxious toils lying, for the most part, useless. He had never been a prosperous, and he was henceforth a disappointed man. A devoted adherent of the Free church of Scotland, his scruples about subscription had debarred him from professional employment in his native country, and his deafness hindered his promotion in the branch he had made peculiarly his own. He did not, however, sink into inaction. Aided by a grant from the Royal Society, he undertook to complete the reduction of the magnetic observations made at the various colonial stations. The task was one of vast and undefined extent, and his sense of responsibility for quarterly payments added anxiety to his labour. His health began to give way, and in 1878 he had a nervous attack, from which he never satisfactorily recovered. A trip to Switzerland produced a partial rally, but on 22 Nov. 1879 he died suddenly, at the age of sixty-two.

His character was a peculiarly estimable one. He united amiability and social charm with rigid integrity and a sensitiveness of conscience ill fitted to advance his material interests. His scientific merits did not receive the cordial recognition they deserved. He took a prominent part in ascertaining the laws of terrestrial magnetism. The discovery is entirely due to him that the earth loses or gains magnetic intensity as a whole in other words, that the changes in the daily mean horizontal force are nearly the same all over the globe. This conclusion, arrived at through a laborious investigation, was first published in a letter to Sir David Brewster, written from Trevandrum on 21 Dec. 1857 (Phil. Mag. xvi. 81, August 1858). In the same communication the existence of a magnetic period of twenty-six days, attributed to the sun's rotation, was announced, and the evidence on both points was detailed in a paper read before the Royal Society of Edinburgh on 4 Feb. 1861 (Trans. R. Soc. Ed. xxii. pt. iii. 511). Independently of, though subsequently to Kreil, Broun deduced from the Makerstoun observations the fact of a lunar-diurnal influence on the declination-needle (Report Brit. Assoc. 1846, ii. 32), a prolonged study of which showed him that it varied in character with the position of the sun (Proc. R. Soc. x. 484, xvi. 59), and in. amount inversely as the cube of the distance of the moon (Trans. R. Soc. Ed. xxvi. 750). He early defined the annual period of magnetic intensity as consisting of a maximum near each solstice, with minima at the equinoxes (Report Brit. Assoc. 1845, ii. 15) ; gave the first complete account of the daily variations of the needle at the magnetic equator (ib. 1860, ii. 21), and reached, in the course of these discussions, the remarkable conclusion that great magnetic disturbances proceed from particular solar meridians.

His researches contributed largely to establish meteorology on a scientific basis. He discovered the 26-day period of atmospheric pressure, showed the wide range of simultaneous barometrical fluctuations, initiated the systematic study of variously elevated cloud-strata, and indicated the connection between atmospheric movements and isobaric lines (Proc. R. Soc. xxv. 515). But he lacked the power of placing his ideas in a striking light, and the independence of his character did not permit him to purchase applause for himself by flattering the opinions of others. The Royal Society admitted him as a member in 1853, and awarded him a royal medal in 1878. His communications to the Royal Society of Edinburgh were honoured with the Keith prize in 1861.

The Royal Society's 'Catalogue of Scientific Papers' enumerates (vols. i. and vii.) fifty-one of his productions, besides which he contributed to the 'Philosophical Transactions' a paper 'On the Variations of the Daily Mean Horizontal Force of the Earth's Magnetism produced by the Sun's Rotation, and the Moon's Synodical and Tropical Eevolutions' (clxvi. 387, 1876) ; to the 'Transactions of the Royal Society of Edinburgh' an elaborate treatise 'On the Decennial Period in the Range and Disturbance of the Diurnal Oscillations of the Magnetic Needle,