In 1903, he was appointed Peter Redpath professor of pure mathematics at McGill University, Montreal, Quebec.
Harkness was for a time a vice President of the American Mathematical Society and associate Editor of its Transactions, was elected a member of the London Mathematical Society and in 1908 became a fellow of the Royal Society of Canada. He published, with Professor Frank Morley, two treatises on the Theory of Functions and collaborated on the article "Elliptic Functions", in the German Encyclopædia of Mathematics (1914–15).
