Many will remember the late Professor J L Burchnall, who became a Fellow of the Royal Society of Edinburgh in 1953, and who, for nearly forty years, was a member of the Mathematics Department at the University of Durham. He occupies a vivid place in the memory of many former students, and also many participants of meetings of the British Mathematical Colloquium. In preparing the following account I have been greatly assisted by his son, Mr H H Burchnall, Registrar of the University of Liverpool, and by Professor T J Willmore, of the University of Durham. I also acknowledge assistance from Mr H E Birkbeck, a former Headmaster of Barnard Castle School, and from Mr G S Cheesbrough, of the Grammar School, Kirkham.

Joseph Langley Burchnell was born on 8 December 1892, in the village of Whichford, in Warwickshire. He was the eldest of six children, four boys and two girls, the family of Henry Walter and Ann Newport Burchnall. His father was the village schoolmaster, but about the turn of the century the family moved to Butterwick, near Boston, in Lincolnshire. Joseph became a pupil at the Boston Grammar school, and held a Holland County Council Scholarship. In his final year, 1911, he gained an Exhibition to Christ Church, Oxford. He was awarded a University Junior Mathematical Exhibition in 1913 and graduated B.A. with first class honours in Mathematics in 1914, just managing thereby to complete his undergraduate education before the outbreak of the First World War. He had a continuing affection both for the Boston Grammar School and for Christ Church, keeping in touch with them throughout his life and remembering them both in his Will.

After the outbreak of war Burchnall was soon in the Army, being commissioned as a Second Lieutenant in the Royal Garrison Artillery in February 1915. He rose to the position of Captain and served in France and Belgium. He was twice wounded in the shoulder in July 1916 and, more seriously, in the leg on 28 March 1918 at the beginning of the German spring offensive of that year. He had to have his right leg amputated and, as a result, was unable to take any further active part in the war. He was awarded the Military Cross in June 1918, and for a short time taught at an Army School for officers in Oxford. He married, 1917, Gertrude Frances Rollinson.

His serious injury was a great handicap for the rest of his life. Although mobile, he had difficulty in getting about and hills, uneven pavements and stairs imposed a considerable restriction. He once remarked that, had he not been crippled, he would have liked to join the Civil Service, but concluded that life in London would have been very difficult. As a result he sought an academic post and in 1919 became a Lecturer in Mathematics at the University of Durham. Later he became a Reader and in 1939 he was appointed Professor of Mathematics. He held this position until 1958, when he retired from Durham. At this time he became an Emeritus Professor and was awarded the O.B.E.

Professor Burchnall, on retirement, went to live at Southwold, Suffolk, but he continued his interest in mathematics and also had wider interests. He was extremely well-read, and it has been said that, had he not become a mathematician, he might well have been an historian. He had a great fund of general knowledge and, to the end of his life, could remember in the greatest detail the places he had visited, the towns he had seen, the roads he had travelled. He died on 29 April 1975 at the age of 82 and is survived by his widow, two sons and one daughter.

Before passing to an appreciation of the mathematical work of Professor Burchnall, it is appropriate to refer to his activity in the social field. For about twenty years he was the representative of Durham University on the Governing Body of Barnard Castle School, and the last of these years saw him 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.

Professor Burchnall had a great aptitude and liking for educational administration. From 1926 to 1938 (when he became Professor of Mathematics) he served, part-time, as secretary to the council of the Durham Colleges. He was thus concerned in the planning and execution of policy at the time of the Royal Commission of 1934 and at the formation of the federal university under the statutes of 1937. He worked increasingly to make the Durham Colleges vigorous, effective and capable of standing on their own, so that he was a main contributor to the strength of independence which made possible the establishment, in 1965, of separate Universities at Durham and at Newcastle upon Tyne. For further details of these matters the reader is referred to the notice by Mr W S Angus, former Registrar of the University of Durham, in The Times on 8 May 1975.

A notable feature of the published work of Professor Burchnall is that about half the papers were written jointly with T W Chaundy. The latter, only three years older than Burchnall, was appointed a lecturer in Mathematics at Christ Church one year after Burchnall went up to that College. The two were close collaborators over a period of twenty years. Their joint work was mostly concerned with the solutions of ordinary differential equations, and the study of special functions. A basic question from the outset concerned differential operators, $P$ and $Q$, each being a polynomial in $D$ ( = $d/dx$) with coefficients which are functions of $x$. Conditions were sought under which $P$ and $Q$ commute. The fundamental theorem, proved by Burchnall and Chaundy, is that if $P$ and $Q$ are of orders $m$ and $n$ respectively, and $PQ = QP$, then $P$ and $Q$ satisfy identically an algebraic equation, $F(P, Q) = 0$, of degree $n$ in $P$ and $m$ in $Q$. Moreover, if $m$ and $n$ are co-prime, then $F(P, Q) = 0$ is a sufficient condition for the identity $PQ - QP = 0$.

At this point the problem develops from that of finding the most general commutative pair of operators to the finding of the general pair satisfying a prescribed identity, $F(P, Q) = 0$. The subject rapidly develops in terms of the geometry of various classes of plane algebraic curves. The theory of multiple points and Abelian equations is involved. It is too extensive to outline here. The substantial paper (3) is that of 1928, although there was an earlier paper (1) and also a later one (8), on the subject. This paper (3), published in the

Apart from his work on commutative operators, Burchnall researched, over a period of about twenty-five years, into the theory of special functions, especially Hypergeometric and Bessel Functions. Most of the results were particular rather than general, and nearly always the guiding light was the study of the differential equations involved. Details will be found in the papers listed in the bibliography, especially the later papers, (12 to 23).

Finally it is noteworthy that several papers appeared on widely differing topics. There is a paper of 1925 (2) on functions discontinuous at rational points, a paper on number theory (Tarry's problem) (11), and a paper on Fourier Transforms (including some skilful use of operator × ( = $x d/dx$) (9).

Professor Burchnall's publications extended over a period of more than thirty years. He was a skilful and versatile mathematician, who found a field in which he could research and stuck to that field throughout his working life. This is somewhat remarkable for one who shone so well at the same time in the field of administration.

1. (With T W Chaundy): Commutative Ordinary Differential Operators:

2. (With T W Chaundy): Functions Discontinuous at Rational Points:

3. (With T W Chaundy): Commutative Ordinary Differential Operators:

4. Functions of an Infinite Number of Variables:

5. (With T W Chaundy): A Set of Differential Equations Which Can be Solved by Polynomials:

6. (With T W Chaundy): A Note on the Hypergeometric and Bessel's Equations:

7. (With T W Chaundy): A Note on the Hypergeometric and Bessel's Equations II:

8. (With T W Chaundy): Commutative Ordinary Differential Operators II: The Identity $P^{n} = Q^{m}$:

9. Symbolic Relations Associated with Fourier Transforms:

10. A Relation Between Hypergeometric Series:

11. (With T W Chaundy): A Type of 'Magic Square' in Tarry's Problem:

12. The Differential Equations of Appell's Function $F_{4}$:

13. (With T W Chaundy): Expansions of Appell's Double Hypergeometric Functions:

14. A Note on the Polynomials of Hermite:

15. (With T W Chaundy): Expansions of Appell's Double Hypergeometric Functions II:

16. Differential Equations Associated with Hypergeometric Functions:

17. The Hypergeometric Identities of Cayley, Orr and Bailey:

18. On the Well-Poised 3_F_2:

19. (With A Larkin): The Theorems of Saalschutz and Dougall:

20. The Bessel Polynomials:

21. An Algebraic Property of the Classical Polynomials:

22. Some Determinants with Hypergeometric Elements:

23. A Method of Evaluating Certain Determinants:

Joseph Langley Burchnell was born on 8 December 1892, in the village of Whichford, in Warwickshire. He was the eldest of six children, four boys and two girls, the family of Henry Walter and Ann Newport Burchnall. His father was the village schoolmaster, but about the turn of the century the family moved to Butterwick, near Boston, in Lincolnshire. Joseph became a pupil at the Boston Grammar school, and held a Holland County Council Scholarship. In his final year, 1911, he gained an Exhibition to Christ Church, Oxford. He was awarded a University Junior Mathematical Exhibition in 1913 and graduated B.A. with first class honours in Mathematics in 1914, just managing thereby to complete his undergraduate education before the outbreak of the First World War. He had a continuing affection both for the Boston Grammar School and for Christ Church, keeping in touch with them throughout his life and remembering them both in his Will.

After the outbreak of war Burchnall was soon in the Army, being commissioned as a Second Lieutenant in the Royal Garrison Artillery in February 1915. He rose to the position of Captain and served in France and Belgium. He was twice wounded in the shoulder in July 1916 and, more seriously, in the leg on 28 March 1918 at the beginning of the German spring offensive of that year. He had to have his right leg amputated and, as a result, was unable to take any further active part in the war. He was awarded the Military Cross in June 1918, and for a short time taught at an Army School for officers in Oxford. He married, 1917, Gertrude Frances Rollinson.

His serious injury was a great handicap for the rest of his life. Although mobile, he had difficulty in getting about and hills, uneven pavements and stairs imposed a considerable restriction. He once remarked that, had he not been crippled, he would have liked to join the Civil Service, but concluded that life in London would have been very difficult. As a result he sought an academic post and in 1919 became a Lecturer in Mathematics at the University of Durham. Later he became a Reader and in 1939 he was appointed Professor of Mathematics. He held this position until 1958, when he retired from Durham. At this time he became an Emeritus Professor and was awarded the O.B.E.

Professor Burchnall, on retirement, went to live at Southwold, Suffolk, but he continued his interest in mathematics and also had wider interests. He was extremely well-read, and it has been said that, had he not become a mathematician, he might well have been an historian. He had a great fund of general knowledge and, to the end of his life, could remember in the greatest detail the places he had visited, the towns he had seen, the roads he had travelled. He died on 29 April 1975 at the age of 82 and is survived by his widow, two sons and one daughter.

Before passing to an appreciation of the mathematical work of Professor Burchnall, it is appropriate to refer to his activity in the social field. For about twenty years he was the representative of Durham University on the Governing Body of Barnard Castle School, and the last of these years saw him 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.

Professor Burchnall had a great aptitude and liking for educational administration. From 1926 to 1938 (when he became Professor of Mathematics) he served, part-time, as secretary to the council of the Durham Colleges. He was thus concerned in the planning and execution of policy at the time of the Royal Commission of 1934 and at the formation of the federal university under the statutes of 1937. He worked increasingly to make the Durham Colleges vigorous, effective and capable of standing on their own, so that he was a main contributor to the strength of independence which made possible the establishment, in 1965, of separate Universities at Durham and at Newcastle upon Tyne. For further details of these matters the reader is referred to the notice by Mr W S Angus, former Registrar of the University of Durham, in The Times on 8 May 1975.

**Mathematical Papers**A notable feature of the published work of Professor Burchnall is that about half the papers were written jointly with T W Chaundy. The latter, only three years older than Burchnall, was appointed a lecturer in Mathematics at Christ Church one year after Burchnall went up to that College. The two were close collaborators over a period of twenty years. Their joint work was mostly concerned with the solutions of ordinary differential equations, and the study of special functions. A basic question from the outset concerned differential operators, $P$ and $Q$, each being a polynomial in $D$ ( = $d/dx$) with coefficients which are functions of $x$. Conditions were sought under which $P$ and $Q$ commute. The fundamental theorem, proved by Burchnall and Chaundy, is that if $P$ and $Q$ are of orders $m$ and $n$ respectively, and $PQ = QP$, then $P$ and $Q$ satisfy identically an algebraic equation, $F(P, Q) = 0$, of degree $n$ in $P$ and $m$ in $Q$. Moreover, if $m$ and $n$ are co-prime, then $F(P, Q) = 0$ is a sufficient condition for the identity $PQ - QP = 0$.

At this point the problem develops from that of finding the most general commutative pair of operators to the finding of the general pair satisfying a prescribed identity, $F(P, Q) = 0$. The subject rapidly develops in terms of the geometry of various classes of plane algebraic curves. The theory of multiple points and Abelian equations is involved. It is too extensive to outline here. The substantial paper (3) is that of 1928, although there was an earlier paper (1) and also a later one (8), on the subject. This paper (3), published in the

*Proceedings of the Royal Society of London,*was immediately followed by one from H F Baker, indicating the intimate connection between the subject and the Abelian Theory. Reference was made to the latter author's great volume of 1897 on 'Abel's Theorem and the Allied Theory', and to the work of Weierstrass in this field. The general conclusion, after reading the papers of Burchnall and Chaundy, and that by Baker, is that the subject of commutative operators is not exhausted, and may well repay further study, perhaps in the light of the more recently developed theory of commutative algebra, which is itself the brainchild of algebraic geometry.Apart from his work on commutative operators, Burchnall researched, over a period of about twenty-five years, into the theory of special functions, especially Hypergeometric and Bessel Functions. Most of the results were particular rather than general, and nearly always the guiding light was the study of the differential equations involved. Details will be found in the papers listed in the bibliography, especially the later papers, (12 to 23).

Finally it is noteworthy that several papers appeared on widely differing topics. There is a paper of 1925 (2) on functions discontinuous at rational points, a paper on number theory (Tarry's problem) (11), and a paper on Fourier Transforms (including some skilful use of operator × ( = $x d/dx$) (9).

Professor Burchnall's publications extended over a period of more than thirty years. He was a skilful and versatile mathematician, who found a field in which he could research and stuck to that field throughout his working life. This is somewhat remarkable for one who shone so well at the same time in the field of administration.

**Bibliography**1. (With T W Chaundy): Commutative Ordinary Differential Operators:

*Proc. Lond. Math. Soc.***21**(1922), 420-40.2. (With T W Chaundy): Functions Discontinuous at Rational Points:

*Proc. Lond. Math. Soc.***24**(1925), 150-157.3. (With T W Chaundy): Commutative Ordinary Differential Operators:

*Proc. Roy. Soc. Lond. A***118**(1928), 557-583.4. Functions of an Infinite Number of Variables:

*Math. Gazette***14**(1929), 513-518.5. (With T W Chaundy): A Set of Differential Equations Which Can be Solved by Polynomials:

*Proc. Lond. Math. Soc.***30**(1929-30), 401-4146. (With T W Chaundy): A Note on the Hypergeometric and Bessel's Equations:

*Quart. J. Math.***1**(1930),186-195.7. (With T W Chaundy): A Note on the Hypergeometric and Bessel's Equations II:

*Quart. J. Math.***2**(1931), 289-297.8. (With T W Chaundy): Commutative Ordinary Differential Operators II: The Identity $P^{n} = Q^{m}$:

*Proc. Roy. Soc. Lond. A***134**(1931), 471-485.9. Symbolic Relations Associated with Fourier Transforms:

*Quart. J. Math.***3**(1932), 213-223.10. A Relation Between Hypergeometric Series:

*Quart. J. Math.***3**(1932), 318-320.11. (With T W Chaundy): A Type of 'Magic Square' in Tarry's Problem:

*Quart. J. Math.***8**(1937), 119-130.12. The Differential Equations of Appell's Function $F_{4}$:

*Quart. J. Math.***10**(1939), 145-150.13. (With T W Chaundy): Expansions of Appell's Double Hypergeometric Functions:

*Quart. J. Math.***11**(1940), 249-270.14. A Note on the Polynomials of Hermite:

*Quart. J. Math.***12**(1941), 9-11.15. (With T W Chaundy): Expansions of Appell's Double Hypergeometric Functions II:

*Quart. J. Math.***12**(1941),112-128.16. Differential Equations Associated with Hypergeometric Functions:

*Quart. J. Math.***13**(1942), 90-106.17. The Hypergeometric Identities of Cayley, Orr and Bailey:

*Proc. Lond. Math. Soc.***50**(1944), 56-74.18. On the Well-Poised 3_F_2:

*J. Lond. Math. Soc.***23**(1948), 253-257.19. (With A Larkin): The Theorems of Saalschutz and Dougall:

*Quart. J. Math. Oxford***1**(1950), 161-164.20. The Bessel Polynomials:

*Canadian J. Math.***3**(1951), 62-68.21. An Algebraic Property of the Classical Polynomials:

*Proc. Lond. Math. Soc.***1**(1951), 232-240.22. Some Determinants with Hypergeometric Elements:

*Quart. J. Math. Oxford***3**(1952), 151-157.23. A Method of Evaluating Certain Determinants:

*Proc. Edinb. Math. Soc.***9**(1954), 100-104.Joseph Langley Burchnall's RSE obituary by L S Goddard appeared in

*Royal Society of Edinburgh Year Book 1976,*30-33.