  We present a systematic investigation of the electrical, structural, and
antiferromagnetic properties for the series of
Ba(Fe$_{1-x-y}$Co$_{x}$Rh$_{y}$)$_{2}$As$_{2}$ compounds with fixed $x \approx$
0.027 and $ 0 \leq y \leq 0.035$. We compare our results for the Co-Rh doped
Ba(Fe$_{1-x-y}$Co$_{x}$Rh$_{y}$)$_{2}$As$_{2}$ compounds with the Co doped
Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ compounds. We demonstrate that the
electrical, structural, antiferromangetic, and superconducting properties of
the Co-Rh doped compounds are similar to the properties of the Co doped
compounds. We find that the overall behaviors of
Ba(Fe$_{1-x-y}$Co$_{x}$Rh$_{y}$)$_{2}$As$_{2}$ and
Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ compounds are very similar when the total
number of extra electrons per Fe/$TM$ ($TM$ = transition metal) site is
considered, which is consistent with the rigid band model. Despite the
similarity, we find that the details of the transitions, for example, the
temperature difference between the structural and antiferromagnetic transition
temperatures and the incommensurability of the antiferromangetic peaks, are
different between Ba(Fe$_{1-x-y}$Co$_{x}$Rh$_{y}$)$_{2}$As$_{2}$ and
Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ compounds.
