Abstract. We study properties of the distribution of the second Ostrogradskiȋ series, for which the differences of terms are independent identically distributed random variables. We completely describe the Lebesgue structure of this distribution. In particular, we prove that it cannot be absolutely continuous. We also develop ergodic theory for the second Ostrogradskiȋ expansion. One of the results is that, for almost all (in the sense of Lebesgue measure) real numbers of the unit interval, an arbitrary symbol of an alphabet occurs finitely often in the corresponding Ostrogradskiȋ difference expansion. We also study properties of the dynamical system generated by the one-sided shift transformations T of the Ostrogradskiȋ difference representation. It is shown that there is no probability measure that is invariant and ergodic with respect to T and absolutely continuous with respect to Lebesgue measure.