Hitherto Thomson's work had lain mainly in pure science ; hut while still engaged on his thermodynamic studies, he was drawn toward the first of those practical applications that made him famous. Early in 1853 he had communicated to the Glasgow Philosophical Society a paper 'On Transient Electric Currents,' in which he investigated mathematically the discharge of a Leyden jar through circuits possessing self-induction as well as resistance. He founded his solution on the equation of energy, ingeniously building up the differential equation and then finding the integral. The result was remarkable. He discovered that a critical relation occurred if the capacity in the circuit was equal to four times the coefficient of self-induction divided by the square of the resistance. If the capacity was less than this the discharge was oscillatory, passing through a series of alternate maxima and minima before dying out.
If the capacity was greater than this the discharge was non-oscillatory, the charge dying out without reversing. This beautiful bit of mathematical analysis passed almost unnoticed at the time, but it laid the foundation of the theory of electric oscillations subsequently studied by Oberbeck, Schiller, Hertz, and Lodge, and forming the basis of wireless telegraphy. Fedderssen in 1859 succeeded in photographing these oscillatory sparks, and sent photographs to Thomson, who with great delight gave an account of them to the Glasgow Philosophical Society.
At the Edinburgh meeting of the British Association in 1854 Thomson read a paper 'On Mechanical Antecedents of Motion, Heat, and Light.' Here, after touching on the source of the sun's heat and the energy of the solar system, Thomson reverted to his favourite argument from Fourier according to which, if traced backwards, there must have been a beginning to which there was no antecedent.
In the same year, in the 'Proceedings of the Royal Society,' appeared the result of Thomson's investigation of cables under the title 'On the Theory of the Electric Telegraph.' Faraday had predicted that there would be retardation of signals in cables owing to the coating of gutta-perclia acting like the glass of a Leyden jar. Forming the required differential equation, and applying Fourier's integration of it, Thomson drew the conclusion that the time required for the current at the distant end to reach a stated fraction of its steady value would be proportional both to the resistance and to the capacity ; and as both of these are proportional to the length of the cable, the retardation would be proportional to the square of the length. This famous law of squares provoked much controversy. It was followed by a further research, 'On Peristaltic Induction of Electric Currents,' communicated to the British Association in 1855, and afterward in more complete mathematical form to the Royal Society.
Submarine telegraphy was now becoming a practical problem of the day [see Bright, Sib Charles Tilstok, Suppl. I]. Sea cables were laid in 1851 between England and France, in 1853 between Holyhead and Howth, and in 1856 across the Gulf of St. Lawrence. In the last year the Atlantic Telegraph Company was formed, with capital mostly subscribed in England, with a view to joining Ireland to Newfoundland. Bright was engineer; Whitehouse (a retired medical practitioner) was electrician ; Thomson (of 2 The College, Glasgow) was included in the list of the directors. In a pamphlet issued by the company in July 1857 it was stated that 'the scientific world is particularly indebted to Professor W. Thomson, of Glasgow, for the attention he has given to the theoretical investigation of the conditions under which electrical currents move in long insulated wires, and Mr. Whitehouse has had the advantage of this gentleman's presence at his experiments, and counsel, upon several occasions.' As a matter of fact Whitehouse had previously questioned Thomson's ’law of squares' at the British Association meeting of 1856, declaring that if it was true Atlantic telegraphy was hopeless. He professed to refute it by experiments. Thomson effectively replied in two letters in the 'Athenæum.' He pointed out that success lay primarily in the adequate section of the conductor, and hinted at a remedy (deduced from Fourier's equations) which he later embodied in the curb signal transmitter. Thomson steadily tested his theories in practice. In December 1856 he described to the Royal Society his device for receiving messages, namely a sort of tangent galvanometer, with copper damper to the suspended needle, the deflections being observed by watching through a reading telescope the image of a scale reflected from the polished side of the magnet or from a small mirror carried by it. Subsequently he abandoned this subjective method for the objective plan in which a spot of light from a lamp is reflected by the mirror upon a scale. It is probably true that the idea of thus using the mirror arose from noticing the reflection of light
