James Thomson | Mathematician
In 1832, James Thomson (mathematician), son of James Thomson (farmer), father of James Thomson (engineer) and William Thompson (physicist), became professor of mathematics at the Glasgow University, a position which he held throughout his life.
From 1836 to 1859, Scottish astronomer John Nichol was a professor of practical astronomy at the University of Glasgow. In this span, for a period of two years, beginning in 1839, Nichol’s taught classes in natural philosophy, as a fill-in for the regular professor William Meikleham who had become absent due to illness. 
When Nichol’s took the chair of natural philosophy, he quickly updated the curriculum, introducing students to the new mathematical works of French mathematical physicist Joseph Fourier's, particularly his 1822 Analytical Theory of Heat. 
The mathematical treatment much impressed young 15-year old William Thomson, who became intrigued with Fourier's Théorie analytique de la chaleur and committed himself to study the “continental” mathematics resisted by a British establishment still working in the shadow of English physicist Isaac Newton.
In particular, after listening to the praises of Fourier’s book by Nichol, likely in one of his lectures, Thomson asked whether he should read the Théorie Analytique, to which Nochol’s replied “the mathematics is very difficult”.  William later got the book from the library. In his own words, “On the 1st of May … I took Fourier out of the University Library; and in a fortnight I had mastered it—gone right through it.”  This was said to be Thomson's first intellectual stimulator as to puzzle behind the nature of heat.
In 1840, Scottish civil engineer Lewis Gordon, a student of the Edinburgh school of thermodynamics, set up a civil engineer partnership with Lawrence Hill in Glasgow. From 1841 to 1855, Gordon was the first holder of the chair of civil engineering and mechanics at Glasgow University. In 1855, supposedly frustrated with the lack of facilities at Glasgow, he handed over his position to William Rankine and moved on to a private practice involved with the designing iron bridges.
During the 1841-42 school year, a 19-year old James Thomson (the older brother) went to Glasgow College and was able to study engineering under Gordon, a newly appointed professor of civil engineering and mechanics (1840-45).  During his mentoring of James, in some way or another, it seems that Gordon's French education at the Ecole Polytechnique, the school of Sadi Carnot, author of Reflections on the Motive Power of Fire (1824) and Emile Clapeyron, author of Memoir on the Motive Power of Fire (1834), the first two papers on thermodynamics, seems to have rubbed off on James, being that a discussion with his younger brother William on the early two French pioneers of heat engine theory (Carnot and Clapeyron). In a letter to William in August, 1844, James inquired who it was that had proved there was a definite quantity of mechanical effect (work) given out during the passage of heat from one body to another. He stated his intentions of writing an article for the Artisan about the theoretical possibility of working steam engines without fuel by using over again the heat which was thrown out in the hot water from the condenser, noting that:
“I shall have to enter on the subject of the paper you mentioned to me”.
This paper, according to historian British energy historian Crosbie Smith, was almost certainly the 1837 translation for Taylor’s Scientific Memoirs of Clapeyron’s “Memoirs on the Motive Power of Heat”.  The discussions in James letter of 1844, according to Smith, are the earliest record of implicit references to Clapeyron and Carnot by the Thomsons.
In 1848, Gordon famously gave William Thomson (the younger brother) a copy of French physicist Sadi Carnot’s 1824 memoir Reflections on the Motive Power of Fire.  It was after reading this memoir, which William had previously spent years searching for (particularly in the bookstores of Paris), that William essentially launched the science of thermodynamics, when he published his 1848 “On an Absolute Thermometric Scale founded on Carnot’s Theory of the Motive Power of Heat, and Calculated from Regnault’s Observations” (written before receiving Carnot's book) and 1849 “An Account of Carnot’s Theory of the Motive Power of Heat; with Numerical Results Deduced from Regnault’s Experiments on Steam” (written after receiving Carnot's book), the latter of which found its way into the hand of German physicist Rudolf Clausius who in turn was so intrigued with this new subject that he "founded", in the retrospect words of American engineer Willard Gibbs, the science of thermodynamics with the publication of his most-famous memoir the 1850 "On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom".
William Thomson, through the workings of his father, James Thomson (mathematician), would later replace Meikleham after his death in 1846 as professor of natural philosophy at Glasgow University, where he was instrumental in the development of and guidance of the Glasgow Philosophical Society. 
In 1849, through the actions of William Thomson, Rankine, a previous student of the University and Edinburgh, was elected a fellow of the Royal Society of Edinburgh, which thus connects him to the Edinburgh school.
Rankine then became professor of civil engineering and mechanics at the University of Glasgow in 1855, as mentioned, until his death in 1872, pursuing engineering research along a number of lines in civil and mechanical engineering. Rankine's vacant position was filled by James Thomson (junior) in 1873 a position he held until his retirement in 1889.
A later thermodynamics publications from this school is the 1892 work of mathematician Peter Alexander. 
1. Biography of John Pringle Nichol – The University of Glasgow Story.
2. Fourier, Joseph. (1822). Analytical Theory of Heat (Théorie Analytique de la Chaleur). Cambridge.
3. Lindley, David. (2004). Degrees Kelvin - a Tale of Genius, Invention, and Tragedy. Washington, D.C.: Joseph Henry Press.
4. Thomson, S.P. (1910). Life of Kelvin, (I: 14). London: MacMillan.
5. McCartney, Mark. (2002). “William Thomson: King of Victorian Physics” Physics World, 02 Dec.
6. Alexander, Peter. (1892). Treatise on Thermodynamics. Glasgow University. Published by Longmans, Green.
● William Thomson (collection) – University of Glasgow Library.
● Biography of John Pringle Nichol – University of Glasgow.