  We construct a manifestly gauge invariant Exact Renormalisation Group (ERG)
whose form is suitable for computation in SU(N) Yang-Mills theory, beyond
one-loop. An effective cutoff is implemented by embedding the physical SU(N)
theory in a spontaneously broken SU(N|N) Yang-Mills theory. To facilitate
computations within this scheme, which proceed at every step without fixing the
gauge, we develop a set of diagrammatic techniques. As an initial test of the
formalism, the one-loop SU(N) Yang-Mills beta-function, beta_1, is computed,
and the standard, universal answer is reproduced. It is recognised that the
computational technique can be greatly simplified. Using these simplifications,
a partial proof is given that, to all orders in perturbation theory, the
explicit dependence of perturbative $\beta$-function coefficients, beta_n, on
certain non-universal elements of the manifestly gauge invariant ERG cancels
out. This partial proof yields an extremely compact, diagrammatic form for the
surviving contributions to arbitrary beta_n, up to a set of terms which are yet
to be dealt with. The validity of the compact expression is reliant on an
unproven assertion at the third loop order and above. Starting from the compact
expression for beta_n, we specialise to beta_2 and explicitly construct the set
of terms yet to be dealt with. From the resulting diagrammatic expression for
beta_2, we extract a numerical coefficient which, in the limit that the
coupling of one of the unphysical regulator fields is tuned to zero, yields the
standard, universal answer. Thus, we have performed the very first two-loop,
continuum calculation in Yang-Mills theory, without fixing the gauge.
