  We perform measurements of the heat capacity as a function of temperature in
URu$_{2-x}$Os$_x$Si$_2$ alloys. Our experimental results show that the critical
temperature of the second-order phase transition increases while the value of
the Sommerfeld coefficient in the ordered state decreases with an increase in
osmium concentration. We also observe the increase in the values of the heat
capacity at the critical temperature as well as a broadening of the critical
fluctuations region with an increase in $x$. We analyze the experimental data
using the Haule-Kotliar model which, in particular, identifies the 'hidden
order' transition in the parent material URu$_2$Si$_2$ as a transition to a
state with nonzero hexadecapole moment. We demonstrate that our experimental
results are consistent with the predictions of that model.
