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.

~ 54-556.

Iigher total differentials.

97

f 54-5.iligher2 f t2of d y
of is equal to f+ y.; the third limiting value is undexa Oy?y- dx'
termined, just as d-x itself is, as long as no law is assigned between
the change of the variable x and that of the variable y; but if such
dy
d2y(.
a dependence exist, we shall have to denote Lim d- by d, (ex. gr.
when the increments of x are in a constant ratio to those of y,
dy: dx =, dy: dx2 0); therefore if =f — ~y x
we have d z 2f +2 - f dy Cf (d y)2 f d2y
~~~xwe have ax — -2 xy dx a c3 - -x y dx2
It is to be remarked, that this expression is not symmetrically
formed relatively to the differentials of x and y; this depends on the
fact that we considered the variable x as independent, and formed
the higher quotients of differences with regard to x (~ 33). It becomes
symmetrical either, when as y varies in proportion to x, d- is to
be considered constant, for then equal changes of y belong to equal
changes of x so that as just stated A dy- 0,and consequently:
d2z 2 f 2 f dy + 2f (dy)2
dx2 a O a by dx a y asdx
or, when the change of x as well as that of y is to be made depend
on a third quantity.
This case we must consider more closely. If x as well as y be
functions of the independent variable t whose change therefore conditions the change of value of x and of y, the differentials dx and
dy are functions of t, multiplied by the differential dt; accordingly
the differential quotient d- is a function of t. Writing dx - cp(t)dt,
dy = b(t) dt, then d — -(t —); if t change and the differential quotient
is to be determined, we have
_\dx _ p (t>(t) - 'q(t) (t).
dt p (t)2
Evidently, in consequence of the equations = - (t), dy = (t)
we can also write: d2 (, () or d2x = (tdt2
dt' = t2p'(t), - 4"(t), q2(t)
d2y -= 4'(t)dt2; introducing these values into the above equation, it
assumes the form
(\dx) _ dx d_ - dy_ _ X (ady\ dxd'y -dyd2x
dt dx2dt or d dx2 - dx
i. e. if in a differential quotient, numerator and denominator are conceived
IHARNACK, Calculus. 7