This is the little-known part of the mathematical history of what we nowadays call the Laplace Transform method of solving differential equations. It is a purely mathematical development of Heaviside's operational methods of electric circuit analysis, which requires of the reader a basic knowledge of differential equations, electric circuit theory, Laplace transforms, some vector analysis, as applied to electromagnetic theory, plus complex variables. I maintain that what we nowadays call the Laplace Transform method ought to be rightly called the Heaviside Transform method, since much of its development was due to Heaviside. The chapter headings are: Differential equations: exponential solutions. The Heaviside Transform. Apportioning magnitudes to the exponential factors. Evaluating U11 - T11, and U12 - T12. Heaviside's Expansion Theorem. Bromwich, and Cauchy's theorem. Applications and extensions. Laplace and difference equations. Note : An example of the computation of mutual energies is available from: home.lizzy.com.au/jeremy.staines/U0r_T0r_example.pdf This was not included in books printed prior to December 2017. A simple computational method for the inversion of Heaviside transforms is also available from home.lizzy.com.au/jeremy.staines/Arithmetical_Inverse_Transforms.pdf Various other articles relevant to the Heaviside Operational Calculus are available from: http://home.lizzy.com.au/jeremy.staines/index.html