Page 80 - Foundations Of Differential Calculus

4

On the Nature of Differentials
of Each Order

112. In the first chapter we saw that if the variable quantity x received an
increment equal to ω, then from this each function of x obtained an incre-
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ment that can be expressed as Pω + Qω + Rω + ··· , and this expression
may be finite or it may go to infinity. Hence the function y, when we write
x + ω for x, takes the following form:
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y = y + Pω + Qω + Rω + Sω + ··· .
When the previous value of y is subtracted, there remains the difference of
the function y, which we express as
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∆y = Pω + Qω + Rω + Sω + ··· .
I
Since the subsequent value of x is x = x + ω, we have the difference
of x, namely, ∆x = ω. The letters P,Q,R,... represent functions of x,
depending on y, which we found in the first chapter.

113. Therefore, with whatever increment ω the variable quantity x is in-
creased, at the same time it is possible to define the increase that accrues to
y, the function of x, provided that we can define the functions P,Q,R,S,...
for any function y. In this chapter, and in all of the analysis of the infinite,
the increment ω by which we let the variable x increase will be infinitely
small, so that it vanishes; that is, it is equal to 0. Hence it is clear that the
increase, or the difference, of the function y will also be infinitely small.
With this hypothesis, each term of the expression
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Pω + Qω + Rω + Sω + ···