The Yang-Yang relation expresses the heat capacity at constant volume, C-V(T,rho), of a fluid linearly in terms of the second temperature derivatives of the pressure and the chemical potential, p double prime>(*) over bar * (T,rho) and mu double prime>(*) over bar * (T,rho). At a gas-liquid critical point C-V diverges so, on approaching T-c from below, either p(sigma)double prime>(*) over bar * (T), or mu (sigma)double prime>(*) over bar * (T), or both must diverge, where the subscript sigma denotes the evaluation of p and mu on the phase boundary or vapor-pressure curve. However, previous theoretical and experimental studies have suggested that mu (sigma)double prime>(*) over bar * (T) always remains finite. To test these inferences, we present an analysis of extensive two-phase heat capacity data for propane recently published by Abdulagatov and co-workers. By careful interpolation in temperature and subsequently making linear, isothermal fits vs specific volume and vs density, we establish that the divergence is shared almost equally between the derivatives p(sigma)double prime>(*) over bar * (T) and mu (sigma)double prime>(*) over bar * (T). A re-examination of the analysis of Gaddy and White for carbon dioxide leads to similar conclusions although the singular contribution from mu (sigma)double prime>(*) over bar * (T) is found to be of opposite sign and probably somewhat smaller than in propane.