When World War I began in 1914, shortly after his graduation, Burchnall joined the army serving in the Royal Garrison Artillery in France and Belgium and rising to the rank of Temporary Captain. He was wounded three times: on two occasions in July 1916 he was wounded in the shoulder, then he lost a leg on 28 March 1918. He was awarded the Military Cross in June 1918:-
For conspicuous gallantry and devotion to duty during an enemy attack. When the detachments had been much reduced by heavy casualties he manned one of the guns with three other officers and kept it in action. Later, he rendered valuable assistance in withdrawing the guns under heavy fire, setting a splendid example of skill and resource.In fact it is fair to say that not only did he show remarkable bravery under military action, but also displayed great bravery with his disability. Nothing seemed more difficult for him and during his whole career he was never heard to complain (and very seldom to even mention) his disability. Unable to continue in active service after being wounded, he was sent to Oxford where he taught at the Army School until 1919. By the end of the war he was married - the wedding to Gertrude Frances Rollinson (born 1892) having taken place in 1917. They had two sons and one daughter.
It is not entirely clear that an academic career was Burchnall's first choice for, later in life, he told a friend that he would have liked to have joined the Civil Service but thought that with his disability life in London would be too difficult. Appointed to Durham University as a lecturer in mathematics in 1919 he remained there for the rest of his career. His influence on the university was marked as the authors of  describe:-
He is reported to have shown his eye for discerning and revealing the truth at his interview in 1919, and his outstanding capacity for academic and financial administration was soon recognised by his colleagues after he joined the staff - by 1925 he was Secretary to the Durham Colleges' Council, a part-time post held jointly with his lectureship. This was an influential position for anyone interested in University politics and administration, as he was; and he enjoyed it for some thirteen years, which was fortunate for Durham because the early thirties brought serious difficulties in the federal University, culminating in the Royal Commission of 1934. Most of the work with the Commission on Durham's behalf fell on his shoulders, and characteristically he did not shrink from it. He played a large part in the discussions on reconstitution, and although the Durham Council's recommendations were not all implemented, it was due to Burchnall and a small group of others that the Durham Colleges emerged from this chastening period with so much of their character intact. The Secretaryship became a full-time post in 1938, and he handed over its responsibilities, but the difficult foundation work had been done - there was a base from which the present healthy structure has been able to develop.From 1939 he occupied the chair of mathematics at Durham, a position he was to hold for 20 years. He served the university in many major administrative capacities, on Court, Senate and almost all major committees. W S Angus writes :-
His acuteness and his practical sense could be relied on to bring discussion to a useful conclusion; as often as not, decision was precipitated by a terse and pithy sentence from him. He saw a little further than most of his colleagues.During World War II he put his military experience from World War I to good use by commanding a section of the Home Guard. He took his role as head of mathematics very seriously and was a strong support to members of staff. He also put much effort into his teaching, as he did with everything he undertook, and was greatly respected by his students who held him in "affectionate awe" :-
No-one ever stayed away from his lectures - an unspoken tribute, not only to the quality of his teaching but also to his fundamental kindness, that must surely have given him considerable pleasure.Almost half of Burchnall's mathematical papers were written with T W Chaundy, with whom he collaborated for 20 years. In both the joint papers and his single author papers he wrote on differential equations, hypergeometric functions and Bessel functions. Formal properties of differential operators are studied in many of his contributions, in particular in his early papers he worked on commuting differential operators. We should also mention his interest in school teaching and, in particular, in the teaching of mathematics. He addressed a conference for school teachers of mathematics in 1929 giving the lecture Functions of an infinite number of variables. This lecture introduced the ideas of functional analysis to school teachers and in it he says that these new ideas are the:-
... natural expression of the analytic spirit of an age which, alike in its religion, philosophy, politics and physics, is constantly digging itself up by the roots to inspect its processes of growth [but] there is in the thought of today a synthetic and perhaps more significant trend.For twenty years he served on the board of governors of Barnard Castle School, latterly as chairman of the governors :-
After the Second World War, development of the school was possible in many directions. This called for wise planning and control and Professor Burchnall was a leading figure at all stages. His service and loyalty to the school will long be remembered.He was elected a fellow of the Royal Society of Edinburgh on 2 March 1953. He had been proposed by Alexander C Aitken, Sir Edmund T Whittaker, Edward T Copson, William L Edge, and I M H Etherington. In 1956 he was awarded an O.B.E. "for services to education and the community". His personality is nicely summed up in  where his role as President of the Staff Cricket team is recounted:-
Indeed his behaviour in that office was a cameo of his personality - he stood alertly as umpire for hours in spite of his disability - he granted appeals against the University secretary and the Chief Constable apparently with identical equanimity - and smiled a little at the efforts of an ageing team to keep pace with undergraduates, but it was the whimsical smile of a man who had the world in perspective - he knew what was happening, just as he knew what he wanted to happen to the University.W S Angus writes :-
His characteristic caution was leavened by perseverance and faith. His humour grew less sardonic and his essential kindliness more manifest as his years advanced and so much that he had worked for came into reality.Burchnall retired in 1959 and, although he continued his interest in mathematics in his retirement, he wrote no further research papers. He left Durham and went to live in Southwold, Suffolk. He had many other interests to occupy him in retirement, for example he served as president of the Old Bostonian Association (associated with Boston Grammar School where he was educated) from 1967 to 1969. He died :-
... peacefully, in an armchair by the fire, reading a book.He was buried in St Edmund's Church Cemetery, Southwold. His wife Gertrude survived him by four years and is buried in the same grave.
Article by: J J O'Connor and E F Robertson