  A short proof is given to the fact that the additional symmetries of the KP
hierarchy defined by their action on pseudodifferential operators, according to
Fuchssteiner-Chen-Lee-Lin-Orlov-Shulman, coincide with those defined by their
action on $\tau$-functions as Sato's B\"acklund transformations. Also a new
simple formula for the generator of additional symmetries is presented.
