He was for a time a Lecturer in Mathematics at the Royal Technical College, Glasgow, then a tutor in mathematics in a tutorial college in Glasgow, directed by Dr R F Muirhead, and finally for many years in charge of that section of Messrs Blackie and Sons, Ltd. dealing with the publication of mathematical books.
Dr Dougall made substantial contributions to both pure and applied mathematics. Discoveries due to him in the theory of Bessel and Mathieu Functions and in potential theory have found their way into the standard treatises on the special functions of mathematical physics. What is known nowadays as "Dougall's theorem" - a result on the sum of a certain generalized hypergeometric series with unit argument-was published in a paper in the Proceedings of the Edinburgh Mathematical Society in 1907.
His main contribution to applied mathematics is in the theory of elasticity, and, in particular, in two long papers published in the Transactions of the Royal Society of Edinburgh in the early years of the century. In these papers Dr Dougall not only solved two of the outstanding problems of the mathematical theory of elasticity, but also developed an analytical method which has had important applications in other branches of applied mathematics. For this work he was awarded the Makdougall-Brisbane Prize for the period 1902-4.
For his contributions to pure and applied mathematics, the University of Glasgow conferred upon him the degree of Doctor of Science.
In his capacity as a reader with Messrs Blackie and Sons, Ltd. Dr Dougall originated and directed a vigorous programme of publishing advanced mathematical books by British authors and translations of mathematical treatises by German and Italian authors. It was due to his initiative that standard works by such authors as Knopp, Courant, Volterra and Levi-Civita were made accessible to English-speaking undergraduates.
Dr Dougall maintained throughout his life an interest in the history of mathematics in Scotland, so that it was natural that he should be invited by the University of Glasgow to give, in 1936, the second of the Gibson Lectures on the History of Mathematics. (The first had been given four years earlier by Albert Einstein.) For that lecture he chose as his subject "James Stirling", a man who had always held a special interest for him since he too had been born in the parish of Kippen.
Dr Dougall was married twice, his second wife pre-deceasing him by only a few weeks. He is survived by a son and two daughters.