(\[ZH\])) from this expression for $Z_{\rm N}$. For example, let us consider the case that $\partial \tilde{G}_{\mbox{\tiny(k)}}/\partial z_{\mbox{\tiny(k)}} \ne 0$. We assume that there are $N_{\mbox{\tiny k}}$ solutions of $\tilde{G}_{\mbox{\tiny(k)}}=0$, $z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}$ $(a_{\mbox{\tiny k}} = 1, 2, \cdots, N_{\mbox{\tiny k}})$, and all of them satisfy the conditions (Eq. (\[xyzG-n\])). Then using the formula for the delta function and the change of variables, Eq. (\[ZN1\]) becomes $$\begin{aligned}
Z_{\rm N} &=& 
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int 
\prod_{{\rm k}=1}^{n} 
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}} 
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}})) 
\left|\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}
{\partial z_{\mbox{\tiny(k)}}}\right|^{-1} e^{-\beta \tilde{H}}
\nonumber \\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} 
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}} 
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}})) 
\left|\frac{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}\right|^{-1}
e^{-\beta \tilde{H}}
\nonumber \\
&=& {\mathcal N}
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H} 
= {\mathcal N} Z_{\rm H}~,
\label{ZH=ZN}\end{aligned}$$ where ${\mathcal N}=\prod_{{\rm k}=1}^{n}N_{\mbox{\tiny k}}$ is a constant normalization factor. This factor is irrelevant to the evaluation of physical quantities.

It is natural to require that $Z_{\rm N}$ should agree with $Z_{\rm H}$ (up to some normalization factor), because we just describe the same physical system using different formulations. It should be noted here that both expressions for $Z_{\rm N}$ (Eq. (\[ZN1\]) or Eq. (\[ZN2\])) are different from that proposed in Ref. 