  We develop the theory of Abelian functions defined using a tetragonal curve
of genus six, discussing in detail the cyclic curve $y^4 = x^5 + \lambda_4x^4 +
\lambda_3x^3 + \lambda_2x^2 + \lambda_1x + \lambda_0$. We construct Abelian
functions using the multivariate $\sigma$-function associated to the curve,
generalising the theory of the Weierstrass $\wp$-function. We demonstrate that
such functions can give a solution to the KP-equation, outlining how a general
class of solutions could be generated using a wider class of curves. We also
present the associated partial differential equations satisfied by the
functions, the solution of the Jacobi Inversion Problem, a power series
expansion for $\sigma(\bu)$ and a new addition formula.
