Page 19 - Foundations Of Differential Calculus

2 1. On Finite Differences
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for x. In a similar way we denote the value of the function by y , y ,
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y ,... , which we obtain when we substitute x +3ω, x +4ω, x +5ω,... .
The correspondence between these values is as follows:
x, x + ω, x +2ω, x +3ω, x +4ω, x +5ω, ...,

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y, y , y , y , y , y , ....
3. Just as the arithmetic series x, x + ω, x +2ω,... can be continued to
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infinity, so the series that depends on the function y: y, y , y ,... can be
continued to infinity, and its nature will depend on the properties of the
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function y.Thusif y = x or y = ax + b, then the series y, y , y ,... is also
arithmetic. If y = a/ (bx + c), the resulting series will be harmonic. Finally,
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if y = a , we will have a geometric series. Furthermore, it is impossible to
find any series that does not arise from some such function. We usually call
such a function of x, because of the series from which it comes, the general
term of that series. Since every series formed according to some rule has a
general term, so conversely, the series arises from some function of x. This
is usually treated at greater length in a discussion of series.
4. Here we will pay special attention to the differences between successive
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terms of the series y, y , y , y ,... . In order that we become familiar with
the nature of differentials, we will use the following notation:
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y − y =∆y, y − y =∆y , y III − y =∆y , ... .
We express the increment by ∆y, which the function y undergoes when we
substitute x + ω for x, where ω takes any value we wish. In the discussion
of series it is usual to take ω = 1, but here it is preferable to leave the value
general, so that it can be arbitrarily increased or decreased. We usually
call this increment ∆y of the function y its difference. This is the amount
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by which the following value y exceeds the original value y, and we al-
ways consider this to be an increment, although frequently it is actually a
decrement, since the value may be negative.
5. Since y II is derived from y, if instead of x we write x +2ω, it is clear
that we obtain the same result also if we first put x + ω for x and then
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again x + ω for x. It follows that y II is derived from y if we write x + ω
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instead of x. We now see that ∆y is the increment of y that we obtain
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when x + ω is substituted for x. Hence, in like manner, ∆y is called the
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difference of y . Likewise, ∆y II is the difference of y , or its increment,
which is obtained by putting x + ω instead of x. Furthermore, ∆y III is
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the difference, or increment, of y , and so forth. With this settled, from
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the series of values of y, namely, y, y , y , y , ... , we obtain a series of
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differences ∆y,∆y ,∆y , ... , which we find by subtracting each term of
the previous series from its successor.