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which may be called the series of the second 

order of differences, and denoted by .

From this series we may proceed to form the series of the third, fourth, fifth, \ldots orders of differences, the general terms of these series being $D_{3}u_{r},D_{4}u_{r},D_{5}u_{r},\ldots$ respectively.

524. Any Required Term of the Series. From the law of formation of the series

it appears that any term in any series is equal to the term immediately preceding it added to the term below it on the left.

Thus $u_{2}=u_{1}+D_{1}u_{1}$, and $Du_{2}=Du_{1}+D_{2}u_{1}$.

By addition, since $u_{2}+Du_{2}=u_{3}$, we have

In an exactly similar manner by using the second, third, and fourth series in place of the first, second, and third, we obtain $Du_{3}=Du_{1}+2D_{2}u_{1}+D_{3}u_{1}$.

By addition, since $u_{3}+Du_{3}=u_{4}$, we have

So far as we have proceeded, the numerical coefficients follow the same law as those of the Binomial Theorem. We shall now prove by induction that this will always be the case. For suppose that

,

then by using the second to the (n +2)th series in the place of the first to the (n+1)th series we have