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 pupilage ended, he returned to Edinburgh and wrote his ‘Experimental Inquiry into the Advantages attending the Use of Cylindrical Wheels on Railways.’ These wheels, although an obvious improvement, never came into use. In 1842–3 he sent various papers to the Institution of Civil Engineers, for which prizes were given. There was one on ‘The Fracture of Axles,’ the conclusions of which led to new methods of construction. In 1844–5 and afterwards until 1848 he was employed under Locke and Errington on various railway projects promoted by the Caledonian Railway Company, of which his father had become secretary.

About 1848 he commenced the series of researches on molecular physics which occupied him at intervals during the rest of his life, and which constitute his chief claim to distinction in the domain of pure science. His first paper on the subject, with the title ‘On an Equation between the Temperature and the Maximum Elasticity of Steam and other Vapours,’ appeared in the ‘Edinburgh New Philosophical Journal’ (1849, xlvii. 28–42), and at the end of that year he sent to the Royal Society of Edinburgh (Journal, xlvii. 235–9) his great paper ‘On a formula for calculating the expansion of liquids by heat.’ He was made a fellow of the Royal Society of Edinburgh in 1849, and awarded the Keith medal in 1854. In July 1850 he read to the British Association at Edinburgh (Report, 1851, pt. ii. pp. 3–6) another paper on a closely connected subject, ‘Elasticity and Heat.’

In 1853 one of his most characteristic papers, ‘On the General Law of the Transformation of Energy,’ was read by him to the Glasgow Philosophical Society (Proceedings, iii. 276–80). In the same year, with James Robert Napier, he projected and patented a new form of air-engine, but the patent was afterwards abandoned. On 2 June 1853 he was elected a fellow of the Royal Society, and sent to that body a great paper on thermo-dynamics, entitled ‘On the Geometrical Representation of the expansive Action of Heat’ (Phil. Trans. 1854, pp. 115–176). From January to April 1855 he lectured in Glasgow University as deputy for Professor Lewis Gordon, on whose resignation he was appointed to the chair of civil engineering and mechanics, 7 Nov. 1855. In 1856 he was created LL.D. of the university of Dublin. In 1856 the preparation of his course of lectures led him to the invention of some remarkable methods connected with ‘Transformation of Structures.’ These are based on the discovery of ‘reciprocal diagrams’ of frames and force, since greatly extended and simplified by Clerk-Maxwell. In 1857 he resigned the associateship of the Institution of Civil Engineers, and shortly afterwards, on the establishment of the Institute of Engineers in Scotland, he was elected the first president. In July 1859 he received a commission as captain in the Glasgow University rifle volunteers, and in 1860, when senior major, commanded the second battalion at the review held by the queen in the Queen's Park, Edinburgh. In 1865 he was appointed consulting engineer to the Highland and Agricultural Society of Scotland, and also became a contributor to the ‘Engineer.’ He communicated valuable matter to the proceedings of the ‘Committee on Designs for Ships of War’ which was appointed after the loss of the Captain, and for the committee calculated the ‘stability of unmasted ships of low freeboard’ and the ‘stability of ships under canvas.’ In May 1872 the value of his professorship was increased by a donation from Mrs. John Elder; but his health was already failing, and he died at 59 St. Vincent Street, Glasgow, on 24 Dec. 1872.

Besides writing in various newspapers, he contributed upwards of one hundred and fifty papers to scientific journals, many of them exhaustive essays on mathematical or physical questions, and genuine contributions to the advancement of science (Catalogue of Scientific Papers, 1871, v. 93–6). The application of the doctrine, that ‘heat and work are convertible,’ to the discovery of new relations among the properties of bodies was made about the same time by three scientific men, William Thomson (afterwards Lord Kelvin), Rankine, and Clausius. Lord Kelvin cleared the way by his account of Carnot's work on the ‘Motive Power of Heat,’ and pointed out the error of Carnot's assumption that heat is a substance and therefore indestructible. Rankine in 1849, and Clausius in 1850, showed the nature of the further modifications which Carnot's theory required. Lord Kelvin in 1851 put the foundations of the theory in the form they have since retained.

Rankine was the author of: 1. ‘On the Means of improving the Water Supply of Glasgow,’ 1852. 2. ‘Mechanical Laws, Formulæ, and Tables,’ 1856, pt. i. (no more published). 3. ‘A Manual of Applied Mechanics,’ 1858; 11th edit. 1885. 4. ‘A Manual of the Steam Engine and other Prime Movers,’ 1859; 13th edit. 1891. 5. ‘A Manual of Civil Engineering,’ 1862; 15th edit. 1885. 6. ‘Useful Rules and Tables relating to Mensuration, Engineering, Structures, and Machines,’ 1866; 7th edit. 1889. 7. ‘Mechanics (Applied),’ 1868. 8. ‘The