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.

60

Successive differ~entiation.

Bk. I. ch. VII.

dy, d2y, d3y, etc., in like manner as the corresponding powers of dx,
are infinitely small respectively of the orders 1, 2, 3, etc. If we wish
to investigate the order in which a function, which is known to
vanish for x = a, is infinitely small at this point, we have to form
the quotient f(x) and to determine for what value of r it remains
(x - a)r
finite. We shall only become possessed of a general method for calculating such 0 quotients by the investigations of next Chapter. It
may be that the order of becoming infinitely small has to be expressed
by a fractional or even irrational number, as, to cite only the simplest,
in case of f(x) = xn at the point x = 0, where n is any positive
number whatever. It is possible even, though we shall only mention
it here, that no number can be found, but only a limit for r, below
which the quotient is zero, above which it is infinitely great. The
simplest example of this kind is f(x) == xa. log (x), (a > 0), in which a
forms such a boundary between the values r.*)
In the applications of the differential calculus to problems of Geometry and Mechanics two courses always present themselves: either
we start from equations between quotients of differences and pass
over from these to differential quotients; or we start from equations
between differences and pass from these to differentials. The
latter frequently corresponds better to the immediate intuition. In this
case we can from the outset facilitate calculation by omitting in
the equation between the quantities still conceived as finite, all the
terms which in the transition to differentials, become infinitely small
of higher order than some term which occurs in the same sum with
them. If for instance y = xn where n6 is a positive integer, then
A y=(x (+Ax)n -xn =,Xn-1i X'+L 1~2Xn- 2AX2+]t3xn-3 a x3-f- Axn;
here as all terms on the right side become infinitely small of higher
order than the first term, the equation Ay == nxn- Ax, though not
exact for finite values, yet expresses for infinitely small ones the
correct value dy =-nxn-ldx. In elementary Stereometry an application of this remark occurs, in proving the theorem for the cubature
of a body bounded by planes: that the volume of a thin slice bounded
by parallel planes can be calculated as that of a prism, provided the
number of the parallel planes becomes infinite. In fact, the volume
of a slice differs from that of a prism with equal base, by a
quantity which is infinitely small of the second order when the
volume of the prism comes to be considered infinitely small of
the first order, on arbitrarily continued subdivision diminishing their
*) Cauchy, Sur les diverses ordres des quantites infiniment petites. Exercices de math6matiques. Tome I.