was seen that they obey Bose statistics. For the expressions (\[pee\]) and (\[en\]) this means that not only the first term, but at least also the second is experimentally observable. One can use the expressions (\[b1l\]) and (\[bl0\]) for a gas of free, ideal bosons to estimate the order of magnitude of the various terms in (\[pee\]) and (\[en\]). One finds that $b_{\ell}^0=
v_0/\ell^{5/2}\lambda^3$, so that $$\begin{aligned}
p&=&\frac{1}{\beta \lambda^3} \sum_{\ell=1}^{\infty}
\frac{e^{-\ell\beta\Delta}}{\ell^{5/2}},
\label{peef}\\
n&=&\frac{1}{\lambda^3} \sum_{\ell=1}^{\infty}
\frac{e^{-\ell\beta\Delta}}{\ell^{3/2}}.
\label{enf}\end{aligned}$$ Using the value $\Delta/k_B=0.71 {\rm K}$ found in [@Goodk], and the maximum temperature $T= 0.85 {\rm K}$ at which the experiments were performed, this shows that the ratio between the second and first terms in (\[enf\]) is $0.15$, and that between the third and first terms is $0.04$.

If experiments can be done that detect the contribution of $15\%$ of the second term in (\[enf\]), it is also possible to detect the effects of the hard-core on the coefficient $b_2$ in (\[en\]). As can be seen from figure \[rat3\], it varies considerably with $\beta t$, and it can even change sign. However, it is difficult to extract this information from the attenuation experiment, since there it is not clear what exactly the relation between 