Heaviside´s proof of his expansion theorem

Heaviside´s proof of his celebrated Expansion Theorem, found scattered in his “Electrical Papers,” is reconstructed. It is based upon his so-called “conjugate theorem” also discovered independently by Routh, which establishes a relation between any two normal modes of oscillation of a dynamical system. Heaviside´s argument applies to systems having finite number of degrees of freedom and no repeated or null roots of the determinantal equation of the system. The relation between Heaviside´s, Carson´s and Wagner´s proofs is also pointed out.