Page 32 - Foundations Of Differential Calculus

1. On Finite Differences 15
I
Let sin x = y. Then y = sin (x + ω), so that
I
∆y = y − y = sin (x + ω) − sin x.
Now,
sin (x + ω) = cos ω · sin x + sin ω · cos x,

4
and we have shown that
2 4 6
ω ω ω
cos ω =1 − + − + ···
1 · 2 1 · 2 · 3 · 4 1 · 2 · 3 · 4 · 5 · 6
and

ω 3 ω 5 ω 7
sin ω = ω − + − + ··· .
1 · 2 · 3 1 · 2 · 3 · 4 · 5 1 · 2 · 3 · 4 · 5 · 6 · 7
When we substitute these series we obtain
ω 2 ω 3 ω 4 ω 5
∆. sin x = ω cos x − sin x − cos x + sin x + cos x − ··· .
2 6 24 120
Example 4. In a unit circle, to find the difference of the cosine of the arc
x.
I
Let y = cos x. Then since y = cos (x + ω), we have
I
y = cos ω cos x − sin ω sin x
and

∆y = cos ω cos x − sin ω sin x − cos x.

From the series referenced above we obtain
ω 2 ω 3 ω 4 ω 5
∆. cos x = −ω sin x − cos x − sin x + cos x − sin x − ··· .
2 6 24 120

22. Since any function of x, which we call y, whether it is algebraic or
transcendental, has a difference of the form

2 3 4
∆y = Pω + Qω + Rω + Sω + ··· ,
if we take the difference again, it is clear that the second difference of y has
the form
3
2
4
2
∆ y = Pω + Qω + Rω + ··· .
4 Introduction, Book I, Chapter VIII; see also note on page 1.