When this Tract was first published in 1937, an important part of it was Chapter VII, containing Liapounoff's classical inequality for the remainder in the Central Limit Theorem, as well as the theory of the related asymptotic expansions. For the Third Edition, this chapter has been partly rewritten, and now brings a proof of the sharper inequality due to Berry and Esseen. Moreover, several minor changes have been made, and the terminology has been somewhat modernized.