3. This is one of the main results of our study, and a strong experimental evidence of a growth of the dynamical correlation length close to T$_{g}$ . We now investigate how our results are related to another estimate of the size of the dynamical heterogeneities, using the temperature derivative of χ$_{1}$ ( ω ). In [7, 8], the T -derivative of the two-body correlation C ( t ) (or of χ$_{1}$ ( ω )) was introduced as a new response function χ$_{T}$ ( t ) = ∂C ( t ) /∂T . It turns out that T χ$_{T}$ ( t ) is also propor- tional to a dynamical correlation volume, but with an unknown, and possibly T dependent prefactor [7, 9]. The line in Fig. 3 is the number of correlated molecules estimated from max$_{ω}$( T ∂ ( χ ′ $_{1}$( ω ) / Δ χ$_{1}$ ) /∂T ) where, due to the unknown value of H , the normalization was chosen such that it coincides with X ( ω ∗ , T = 202 K ). We see that the T dependencies of these two quantities are close. However, whereas the increase of X ( ω ∗ , T ) is a proof of a growing correlation length, one needs some extra assumptions to infer this from χ$_{T}$ [7]. That χ$_{3}$ and χ$_{T}$ lead to a similar T dependence for N$_{corr}$ ( T ) suggests that these assumptions are indeed warranted (at least for glycerol) and validates the simpler procedure advocated in [6], and used extensively in [8], to extract N$_{corr}$ ( T ). A precise identity relating χ$_{T}$ and χ$_{3}$ has been obtained in [13]. It holds at low enough ω when the linear response depends only on external parameters ( T , density, electric field, . . . ) via the dependence of τ ( T ) on these parameters, a property called time temperature superposition (TTS), and reads: where κ is independent of ω and ˆ χ$_{T}$ is the Fourier transform of χ$_{T}$ . Corrections to this relation in the regime ωτ ≪ 1 are due to weak violations of TTS and to the additional term that corresponds to ω independent non-cooperative degrees of freedom. The continuous line on Fig. 4 presents X ( ω, T ) calculated from the linear susceptibility, according to Eq. (2) with κ = 7 . 6 × 10 − 17 K(m/V) $^{2}$. This value of κ was chosen to be consistent with Fig. 3, i.e. by demanding that the maximum over ω of κ | ˆ χ$_{T}$ (2 ω ) | corresponds to X = 0 . 40 which is the value of the continuous line of Fig. 3 for T = 210 . 2 K . Although Eq. (2) should only be valid at small ω , we see that it reproduces qualitatively the overall behavior of the data at 210 . 2 K. At low frequency, ωτ < 0 . 2, the discrepancy can be understood by the contribution of the ω independent trivial saturation effect noted above. For ωτ > 1, our χ$_{3}$ ( ω ) ≈ κ ˆ χ$_{T}$ (2 ω ) ωτ ≪ 1 , (2) 7