Fox only remained at Imperial College for one year before being appointed as a Lecturer in Mathematics in Birkbeck College, part of the University of London, in 1920. He had been appointed to this lectureship without having any mathematical publications and in fact he did not submit his first paper for publication until 1925. This paper, A class of null series, was published in the Proceedings of the London Mathematical Society in 1926. He begins the paper as follows:-
Various Schlömilch series, representing null functions, have been discovered by Nielsen , who gave proofs of them by arguments similar to those by which the Riemann-Lebesgue lemma is proved. The methods of contour integration, however, give extremely simple proofs of these results, and also give rise to many interesting results which, I believe, are new.He submitted two further papers in June 1925, The Expression of Hypergeometric Series in Terms of Similar Series, and Some Further Contributions to the Theory of Null Series and Their Connexion with Null Integrals to the same Proceedings; both were published in 1927 as was his next paper A Generalization of an Integral Equation Due to Bateman which he submitted in 1926. Fox had useful discussions with George Jeffery who was the professor of Pure Mathematics at University College, London, and in his next paper The Asymptotic Expansion of Generalized Hypergeometric Functions (1928) he thanked George Jeffery for his assistance. Fox was awarded a D.Sc. by the University of London in 1928.
In 1932 he married Eileen Kaye in London; they had a son, Edward, and a daughter, Frances. He emigrated to Canada in 1949 to take up an appointment at McGill University. In 1956 he was promoted to professor at McGill, a post he held until he retired in 1967. His next eight years were spent as visiting professor at Sir George Williams University (now Concordia University) Montreal where he continued to teach mathematics until he was close to 80 years of age.
Fox's main contributions were on hypergeometric functions, integral transforms, integral equations, the theory of statistical distributions, and the mathematics of navigation. In the theory of special functions he introduced an H-function with a formal definition. It is a type of generalisation of a hypergeometric function and related ideas can be found in the work of Salvatore Pincherle, Hjalmar Mellin, Bill Ferrar, Salomon Bochner and others. He wrote only one book An introduction to the calculus of variations (1950, 2nd edition 1963, reprinted 1987). He says in the Preface that he wrote it because:-
During my many years of teaching at London University I felt that none of the existing texts covered the subject as I would like to teach it and so I undertook the task of writing one of my own.How many texts have been written for this reason! Peter Hilton, in a review of the book, writes :-
On the flyleaf of this book it is claimed that "In this work the Calculus of Variations is developed both for its intrinsic interest and because of its wide and powerful applications to modern Mathematical Physics." It appears, however, that the author's main preoccupation is with the applications and that his interest in the Calculus of Variations derives from its applicability to physical problems rather than its intrinsic discipline. ... The book provides an excellent introduction to the subject for those primarily concerned with having available techniques and rules of procedure for tackling concrete problems, and a most exhaustive treatment of the classical problems of the Calculus of Variations (e.g. the brachistochrone, the surface of minimum resistance, the principle of least action) is among the best features of the book. However, for the student of pure mathematics, it does not seem to replace Hadamard's 'Leçons sur le Calcul des Variations' or Bliss's 'Calculus of Variations' (whose mathematical content broadly coincides with its own).L M Graves, however, gives a much more critical review:-
This book, designed as a text for undergraduate students, includes a large number of examples, and devotes chapters 5, 6, and 7 to applications to mechanics, relativity, and elasticity. Unfortunately, the meaning of terms used is often very vaguely conveyed, and proofs are often very unsatisfactory. The errors and defects are too numerous to mention in detail.Among the honours that Fox received, we mention in particular that he was elected a Fellow of the Royal Society of Canada in 1961 and was awarded an honorary LL.D. degree by Concordia University in 1976.
Let us end this biography with a quotation from :-
Charles Fox leaves behind him the memory of a quiet family man, an effective teacher of mathematics who made significant contributions to his field of expertise, and who inspired many of his colleagues and students to carry out independent researches for themselves. He will be remembered for his great intellectual gifts and research contributions, for his courtesy and kindliness and, above all, for his qualities of honesty and integrity.
Article by: J J O'Connor and E F Robertson