  We take the manifestly gauge invariant exact renormalisation group previously
used to compute the one-loop beta function in SU(N) Yang-Mills without gauge
fixing, and generalise it so that it can be renormalised straightforwardly at
any loop order. The diagrammatic computational method is developed to cope with
general group theory structures, and new methods are introduced to increase its
power, so that much more can be done simply by manipulating diagrams. The new
methods allow the standard two-loop beta function coefficient for SU(N)
Yang-Mills to be computed, for the first time without fixing the gauge or
specifying the details of the regularisation scheme.
