Substituting Eqs.\ref{eq:HeatCapacity} and \ref{eq:MaxwellRelation} into Eq. \ref{eq:Substitution}, we have
\begin{equation} \label{eq:Substitution2} \label{eq:Substitution2}-C_{H}dT-T\left(\frac{\partial M}{\partial T}\right)_{H}dH=\kappa\Delta Tdt+C_{\textrm{addenda }}dT\\ \end{equation}dividing through by \(\kappa\ dT\), and setting \(C=C_{H}+C_{\textrm{addenda }}\), we conclude
\begin{equation} \label{eq:SweptFieldMCE} \label{eq:SweptFieldMCE}-\frac{T}{\kappa}\left(\frac{\partial M}{\partial T}\right)_{H}\frac{dH}{dt}=\Delta T+\frac{C}{\kappa}\frac{dT}{dt}\\ \end{equation}as in reference \cite{Fortune_2009}. Note here that \(dT/dt=\frac{d}{dt}(\Delta T)\) since the reservoir temperature is held constant.