Page 22 - Foundations Of Differential Calculus

1. On Finite Differences 5
in like manner,

4 IV III II I
∆ y = y − 4y +6y − 4y + y
and
5 V IV III II I
∆ y = y − 5y +10y − 10y +5y − y.
We observe that the numerical coefficients of these formulas are the same as
those of the binomial expansion. Insofar as the first difference is determined
III
I
II
by the first two terms of the series y, y , y , y , ... , the second difference
is determined by three terms, the third is determined by four terms, and
so forth. It follows that when we know the differences of all orders of y,
II
I
likewise, differences of all orders of y , y , ... are defined.
11. It follows that for any function, with any values of x and any differ-
ences ω, we can find its first difference as well as its higher differences. Nor
is it necessary to compute more terms of the series of the values of y, since
we obtain the first difference ∆y when for the function y we substitute
I
x + ω for x and from this value y we subtract the function y. Likewise the
second difference ∆∆y is obtained from the first difference ∆y by substi-
I I
tuting x + ω for x to obtain ∆y , and then subtracting ∆y from ∆y .In
3
a similar way we get the third difference ∆ y from the second difference
∆∆y by putting x + ω for x and then subtracting. In the same way we
4
obtain the fourth difference ∆ y and so forth. Provided that we know the
first difference of any function, we can find the second, third, and all of
the following differences, since the second difference of y is nothing but the
first difference of the first difference ∆y, and the third difference is nothing
but the first difference of the second difference ∆∆y, and so forth.

12. If a function y is the sum of two or more functions, as for example
I
I
I
I
y = p+q +r +··· , then, since y = p +q +r +··· , we have the difference
∆y =∆p +∆q +∆r + ··· .
Likewise,
∆∆y =∆∆p +∆∆q +∆∆r + ··· .

It follows that if a function is the sum of other functions, then the com-
putation of its differences is just as easy. However, if the function y is the
product of two functions p and q, then, since
I I I
y = p q

and
I
p = p +∆p