A rigorous mathematical description is given for the physical concept of the cascade process of sequential breakdown of turbulent eddies. It is assumed that the probability distribution for the ratio of typical values of turbulent fluctuations averaged over a small volume v to those averaged over a surrounding larger volume V is invariant under the group of space similarity transformations, provided that the length scales of both volumes are much smaller than the external scale of turbulence and much larger than the Kolmogoroff scale. This assumption is in general agreement with the conclusions of Kolmogoroff and Obukhov and with the existing measurements of the power spectrum for the square of the velocity derivative. Some measurements of probability distributions for smallscale turbulent fluctuations are also discussed.

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