An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 32 —34.

Different orders of infinitely small quantities.

59

therefore
Ax - Ax AIx = A2, 2 xA - A2x = A3x 0
Ax — Ax1 A2 x 0,= Ax - A2x-= A3xI ==0
Ax3 - Ax2 = A2x2 = 0.....
Thus the higher differences of the independent variable vanish,
since its values are supposed to increase by equal amounts.
The same is the case, when the function y increases in proportion
to x, that is, when y ax- +b.
Since according to the above method of determination the differential quotients
_?, (- dy d, 'y _ ]/....
dy () 2 n f (x) d - " (x) etc.
have the signification of actual fractions, we can pass over from them
also to the equations between the differentials:
dy=f'(x).dx, d12y- -f"(x).dx2, cd3y=f"'(x).dx3,.. y- f()..dny=. dx.
Of course we have now on each side of such an equation a vanishing
quantity, so that it appears not to contain anything more than
the self-evident identity 0 = 0. Nevertheless it has a determinate
content when we recollect how it originated. For it then asserts:
the nth difference Any at the point x is more nearly equal to the
product of Axn by the determinate value f"(x), the smaller Ax is
chosen; so that the limiting value of the quotient n-Y which we
A Xn
have denoted by d Yis equal to fn(x). f Thus an equation between ind xn
finitely small quantities has a determinate content, if it can be interpreted as a relation between the limiting values of continuous variables.
Now it is further to be remarked, that in the above equations the
differential dx occurs in increasing powers, so that we are enabled
to distinguish infinitely small quantities of different orders. If dx be
called infinitely small of the first order, then dx2 is infinitely small
of the second, dx3 of the third, dxn of the nth order. The ratio of two
infinitely smalls of the nth order and of the mnth order (n > m) is itself
infinitely small of the (n - m)th order: dxn: dxn -= dxn-m. The
numerator may be said to converge to zero much more rapidly than
the denominator; if n == m then the quotient is finite, equal to 1.
34. This measure of becoming infinitely small can be stated generally: Two quantities are infinitely small of the same order when their
quotient retains a finite value. The derived functions
f'(x), f"(x), f"'(x) etc.
have in general for any x finite values; consequently the differentials