Paul Mesham was awarded the degree Master of Science (MSc) by the University of Liverpool in 1912. That same year he was appointed as lecturer in physics at the Natal University College, Pietermaritzburg, and admitted to the MSc degree of the University of the Cape of Good Hope on the basis of his degree from the University of Liverpool. Some five or six years later he became the first professor of physics at the college, a post he held until 1947.
During his entire career Mesham published only a few papers, including the following: 'Roentgen radiation from substances of low atomic weight' (with C.A. Sadler; The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 1912; also in Journal of the Roentgen Society, 1913); and 'Electric charges on bacteria' (South African Journal of Science, 1927, Vol. 24, pp. 121-135). He became a member of the South African Association for the Advancement of Science in 1914 and served as joint secretary of Section A (which included physics) at the association's annual congress in Pietermaritzburg in 1916.
Mesham was married to Dr Nellie Wall.