Page 34 - Foundations Of Differential Calculus

1. On Finite Differences 17
all, proceeding backwards, if we have found the difference for some function
and that difference is now given, we can, in turn, exhibit that function
from which the difference came. Thus, since the difference of the function
ax + b is aω, if we are asked for the function whose difference is aω,wecan
immediately reply that the function is ax + b, since the constant quantity
b does not appear in the difference, so we are free to choose any value for
b. It is always the case that if the difference of a function P is Q, then the
function P + A, where A is any constant, also has Q as its difference. It
follows that if this difference Q is given, a function from which this came is
P +A. Since A is arbitrary, the function does not have a determined value.
25. We call that desired function, whose difference is given, the sum. This
name is appropriate, since a sum is the operation inverse to difference, but
also since the desired function really is the sum of all of the antecedent
values of the difference. Just as
I
y = y +∆y
and
I
II
y = y +∆y +∆y ,
if the values of y are continued backwards in such a way that what x − ω
corresponds to is written as y , and y II preceding this, and also y , y ,
IV
I
III
y ,... , and if we form the retrograde series with their differences
V
y , y , y , y , y , y
V
IV
III
II
I
and
∆y , ∆y , ∆y , ∆y , ∆y ,
IV
I
III
II
V
then y =∆y + y . Since y =∆y + y and y =∆y III + y ,wehave
II
II
III
I
I
II
I
y =∆y +∆y +∆y +∆y +∆y + ··· .
I II III IV V
Thus the function y, whose difference is ∆y, is the sum of the values of
the antecedent differences, which we obtain when instead of x we write the
antecedent values x − ω, x − 2ω, x − 3ω, ... .
26. Just as we used the symbol ∆ to signify a difference, so we use the
symbol Σ to indicate a sum. For example, if z is the difference of the
function y, then ∆y = z. We have previously discussed how to find the
difference z if y is given. However, if z is given and we want to find its
sum y,welet y =Σz, and from the equation z =∆y, working backwards,
we obtain the equation y =Σz, where an arbitrary constant can be added
for the reason already discussed. From the equation z =∆y, if we invert,
we also obtain y =Σz + C. Now, since the difference of ay is a∆y = az,