analyzed as a sum of an electronic $C_{e}$ ($C_{S}$ when superconductivity is to be emphasized), a Debye $C_{D}$ $(=\beta T^{3})$, a nuclear $C_{N}$, and a magnetic contribution $C_{M}$ from the only magnetically active $R$-sublattice. At temperatures of interest, $C_{e}$ and $C_{D}$ were estimated based on our specific heat characterization [@Takeya-YNi2B2C] of single crystal YNi$_{2}$B$_{2}$C ($\gamma=17.5$ mJ/moleK$^{2}$ and $\beta=$ 0.12 mJ/moleK$^{4}$) which had been synthesized by the very same procedures as the one used for the other single crystals. Thus for each compound, $\gamma$ was taken to be the same value as that of YNi$_{2}$B$_{2}$C while $\beta$ was evaluated by normalizing the $\beta$ of YNi$_{2}$B$_{2}$C using the approximation $\beta\propto m^{\frac{3}{2}}$(see Tab.II).

Within the superconducting region, $C_{S}$ was evaluated as 3$\gamma
T^{3}/T_{c}^{2}$. At any rate, for all the studied compounds, $C_{M}(T)$ is much larger than the sum of $C_{e}$ and $C_{D}$. Consequently, even if $C_{e}$ and $C_{D}$ are taken as the bare values of YNi$_{2}$B$_{2}$C, $C_{M}(T)$ would not be noticeably modified, ensuring that our conclusions would not be influenced.

$C_{N}(T)$ of $R$Ni$_{2}$B$_{2}$C, when available, is of  dominant importance only at very low temperatures and was evaluated by least-square fit using the appropriate hyperfine Hamiltonian; the obtained parameters (shown in Tab.II) compare favorably 