Prior to Heaviside’s formulation in 1876. the first steps to developing transmission line theory had already been taken by William Thomsen, otherwise known by Lord Kelvin. He first proposed the idea of the lumped element model, stating ‘the quantity of electricity on a length dx of wire … will be v*c*dx,” and also formulated that signal propagation through transmission lines behaved very similarly to the dispersion of heat in the heat equation meaning that he could simply use the solution already derived by Fourier (using Fourier series to model any function as a sum of sine functions, which over time attenuated to a factor of itself) to model the propagation voltage and current of telegraph signals. 
Additionally, using Fourier’s solution, he formulated higher frequency signals would attenuate faster, and that the time delay is proportional to kcl^2, where K is resistance, C is capacitance and L is length. This means “the time required for each deflection will be sixteen times as long as would be with a wire a quarter of the length.” To combat this, Thomsen proposed a larger cable to have a lower resistance and capacitance, but in the case of the first transatlantic telegraph cable in 1858, was largely ignored, instead opting for Whitehouse’s recommendation of a thinner cable. This led to much longer transmission times, and additionally following Whitehouse, who did not believe in Thomson’s theory, sent over 2000V through the cable, destroying it indefinitely.
The next cable in 1866 followed Thomson’s recommendations and following its success, Thomson was knighted to become Lord Kelvin
Heaviside however, added onto Kelvin’s work by including the idea of inductance, with a proportional voltage drop along the line not only proportional to the rate of change of the charge, but also the rate of change of the current.
While Thomsen introduced the idea of diffusion, Heaviside introduced the ideas of wave dispersion. By doing so, he noticed the “extra current” caused by the inductance of the transmission line
Solution of lossy telegraphers' equation with dispersion vs lossless wave-equation
Here you can see the equations written out by Heaviside in his Electrica Papers, and the modern format of the equations.
XIX. On the extra current – Oliver Heaviside
III. On the theory of the electric telegraph – William Thomsen
Electrical papers : Heaviside, Oliver, 1850-1925 : Volume II
On Heaviside’s contributions to transmission line theory: waves, diffusion and energy flux