In the same year he became a lecturer in mathematics, Barnes was ordained deacon by the Bishop of London and from 1906 to 1908 was Junior Dean of Trinity. In 1915, Barnes left Cambridge, and his career as a professional mathematician, upon his appointment as Master of the Temple in London. This was followed in 1918 to a Canonry of Westminster and finally, in 1924, to the Bishopric of Birmingham, an office he held until 1952 when he had to retire on account of ill-health. He died at his home in Sussex at the age of 79, survived by his wife and two sons.
Barnes' episcopate was marked by a series of controversies stemming from his outspoken views and, rather surprisingly for someone who held such high office in the Church, often unorthodox religious beliefs. In 1940 he lost a libel case in which he had attacked the Cement Makers' Federation for allegedly holding up the supply of cement, for their own profit, at a time of great national need in the construction of air-raid shelters. Undaunted by this set-back, Barnes returned to his accusations on the cement ring in a speech he delivered in the House of Lords the following year, in which he claimed that powerful business concerns were using libel and slander action to suppress criticism. As a theological author, Barnes' book in 1947, entitled The Rise of Christianity, aroused such fierce opposition and criticism from more orthodox members of the Church, that it was strongly suggested he should renounce his episcopal office, a hint which Barnes did not take.
In all, Barnes wrote 29 mathematical papers during the years 1897-1910. His early work was concerned with various aspects of the gamma function, including generalisations of this function given by the so-called Barnes G-function, which satisfies the equation
hypergeometric functions. In the last five of his papers dealing with the hypergeometric functions, Barnes made extensive use of the integrals studied by Mellin in which the integral
involves gamma functions of the variable of integration. It was in these papers that he brought to the attention of British mathematicians the power and simplicity associated with these integrals, and which now bear the name Mellin-Barnes integrals. His last mathematical paper, published in 1910, was a short and elegant demonstration of a previously known result of Thomae concerning a transformation of a generalised hypergeometric function of unit argument into a more rapidly convergent function of the same kind.
Article by: R Paris, University of Abertay, Scotland