  It is suggested that microscopic vortex rings (MVR) play an important role in
the \lambda-transition in helium-II and substantially determine the value of
T_{\lambda}. For very thin films of He-II, with thickness d less than the size
of the smallest MVR, the rings do not fit in and, therefore, do not exist in
such films. Consequently, for superfluid films of He-II, a peculiarity in the
form of a smoothed-out jump should be observed in the curve T_{m}(d) at the
values of thickness approximately equal to the size of the smallest MVR, d= 3 -
9 A (T_{m} is the temperature of the maximum of the broad peak on the curve of
the dependence of the specific heat on temperature). The absence of a similar
peculiarity will be an evidence that MVR do not influence the values of
T_{\lambda} and T_{m}, and do not play any key role in the \lambda-transition.
The currently available experimental data are insufficient for revealing the
predicted peculiarity.
