In November 1839, he entered the University of Edinburgh with a view to becoming a civil engineer, and studied under Philip Kelland, James David Forbes, and Robert Jameson.
[1] In December 1842, Sir Thomas Makdougall-Brisbane - on the advice of Forbes - engaged Welsh as an observer at his observatory at Makerstoun under then director, John Allan Broun.
In 1850 Welsh - who wished to move on - was recommended by Brisbane to William Henry Sykes, chairman of the committee of the British Association which managed Kew Observatory.
In March and May 1854 he made for the committee an investigation on the "pumping" of marine barometers, an artefact of the rolling motion of ships.
In the same year Welsh was directed to construct self-recording magnetic instruments on models devised originally by Ronalds and improved by himself.
[1] At the Ipswich meeting of the British Association in October 1851, Welsh read a long report on Francis Ronalds's three magnetographs.