  We have investigated the elastic properties of the Ce-based filled
skutterudite antimonides CeRu$_{4}$Sb$_{12}$ and CeOs$_{4}$Sb$_{12}$ by means
of ultrasonic measurements. CeRu$_{4}$Sb$_{12}$ shows a slight increase around
130 K in the temperature dependence of the elastic constants $C$$_{11}$,
($C$$_{11}$-$C$$_{12}$)/2 and $C$$_{44}$. No apparent softening toward low
temperature due to a quadrupolar response of the 4$f$-electronic ground state
of the Ce ion was observed at low temperatures. In contrast CeOs$_{4}$Sb$_{12}$
shows a pronounced elastic softening toward low temperature in the longitudinal
$C$$_{11}$ as a function of temperature ($T$) below about 15 K, while a slight
elastic softening was observed in the transverse $C$$_{44}$ below about 1.5 K.
Furthermore, CeOs$_{4}$Sb$_{12}$ shows a steep decrease around a phase
transition temperature of 0.9 K in both $C$$_{11}$ and$ C$$_{44}$. The elastic
softening observed in $C$$_{11}$ below about 15 K cannot be explained
reasonably only by the crystalline electric field effect. It is most likely to
be responsible for the coupling between the elastic strain and the
quasiparticle band with a small energy gap in the vicinity of Fermi level. The
elastic properties and the 4$f$ ground state of Ce ions in CeRu$_{4}$Sb$_{12}$
and CeOs$_{4}$Sb$_{12}$ are discussed from the viewpoint of the crystalline
electric field effect and the band structure in the vicinity of Fermi level.
