  The resonant period drop observed at low temperatures in torsional
oscillators containing solid helium had been interpreted as a signature of
supersolid. However, it was found that the shear modulus increase found in
solid helium at the same low temperature could also decrease the resonant
period of the torsional oscillator. We report the results of a study in two
different torsional oscillators that were specially designed to minimize the
shear modulus effect and maximize any possible supersolid response. We were
able to place an upper limit on the nonclassical rotational inertia or
supersolid fraction of $4\times10^{-6}$. Moreover, we have repeated an earlier
experiment on hcp $^3$He solid, which shows similar low temperature stiffening
as in hcp $^4$He. We found that the small drop of the resonant period measured
in the hcp $^3$He sample is comparable in size to that found in the hcp $^4$He
samples. These results strongly suggest that the resonant period drop reported
in most torsional oscillator studies in the last decade is primarily a
consequence of the shear modulus stiffening effect.

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