the cluster spectrum as a function of length we can then ask what is the spectrum when re-expressed as a function of the radius (of gyration) of the cluster. In [@hart98] we obtained this spectrum by determining the average radius as a function of length, and folding that in with the number density as a function of length. This is an approximate procedure (forced upon us by the fact that we did not foresee the interest of this spectrum during the processing of the clusters) and one can obtain the spectrum more accurately by calculating $r$ for each cluster and forming the spectrum directly. Doing so for the $L = 32$ lattice at $\beta =
2.5115$, also in Figure \[fig\_len\_spec\], we find a power law as in eqn. \[eqn\_rad\_spec\] with $\eta = 4.20$ (8). The spectrum on the $L=20$ lattice at $\beta=2.5$ yields $\eta = 4.27$ (6). We recall that in [@hart98] we claimed that the spectrum was consistent with the scale invariant result $dr/r \times 1/r^4$, [*i.e.*]{} $\eta = 5$. This followed from the fact that we found the the radius of the smaller clusters to vary with their length as $r(l) = s + t.l^{0.5}$ [*i.e.*]{} just what 