  We present a new systematic method to construct Abelian functions on Jacobian
varieties of plane, algebraic curves. The main tool used is a symmetric
generalisation of the bilinear operator defined in the work of Baker and
Hirota. We give explicit formulae for the multiple applications of the
operators, use them to define infinite sequences of Abelian functions of a
prescribed pole structure and deduce the key properties of these functions. We
apply the theory on the two canonical curves of genus three, presenting new
explicit examples of vector space bases of Abelian functions. These reveal
previously unseen similarities between the theories of functions associated to
curves of the same genus.
