Introduction. In an earlier paper 1 BergströM (1) View all notes I proved the inequality for the difference between the normal d. f. 2 Distribution function View all notes Φ (χ) and the d. f. of the sum of n equally distributed random variables with the mean value O. Here σ denotes the dispersion, β3 the absolute third moment of the variable Xi and C is an absolute constant. To establish the inequality I gave an identical expansion of the convolution , when the dispersion for F(χ) was 1, and a lemma for Weierstrass' singular integral. I also remarked that this method could be used for d. f.'s in the space Rk, k> 1. In fact there is very little to be changed when I now give the generalization for the space Rk.