**General Mathematics, Vol. 4, 1996 **

**Abstract:** In the book entitled "Methodus differentialis" published in
1711, Newton indicates in the cases $m\in \{5,6\}$, without proofs, new formulas for his
interpolation polynomial $N_{m}$ introduced in 1687. Arnold Kowalewski [1] has proved
these formulas only in these particular cases. Slight modifications of these proofs are
published in 1932 by Gerhard Kowalewski [2]. At the same time, in [2] it is shown that the
new formulas are more efficient in the computation than the original formula of Isaac
Newton. The aim of this paper is to establish the interpolation formulas indicated in
particular cases by A. and G. Kowalewski brothers, for an arbitrary $m$.

**Classification (MSC91): 41A05**

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