COR. Suppose that our object is to change the independent variable from a to y: then, by the formulæ (4), (5), (6), considering d'y constant, and therefore equating dry, d'3y,....to zero, we have

to change the independent variable from x to 0, when = cos 0.

In this case d'y = f'(x) dx = -ƒ'(x) sin 0 d0; differentiating again, considering de constant, we have

d'y = − ƒ"(x) dx sin 0 d0 – ƒ'(x) cos 0 do2

In this case

or

d'y = f'(x) dx = ƒ'(x) eo d0,

d'2y = ƒ"(x) dx eo d0 + ƒ'(x) eo d03

= ƒ'(x) e2o d02 + ƒ'(x) e® d03,

d"3y = ƒ""(x) dx e2o d02 + 2ƒ“ (x) e2o d03 +ƒ'(x) dx eo d02+ƒ'(x) eo d03

= ƒ"(x) e3o d03 + 3ƒ“(x) e2o d✪3 + ƒ'(x) e° d03,

which reduces the proposed equation to the form

,

Order of Partial Differentiations indifferent.

48. The following is a theorem of great importance in successive differentiation

if

§‚1⁄2§Â‚μ = {ƒ(y, + dy1, Y2 + dy2) − ƒ (Y1, Y2 + dy2)}

− {ƒ(y1 + dy1, Y2) — ƒ(Y1, Y2)}

-

In precisely the same way it may be shewn that

§ï‚§Â ̧u=ƒ(y2+ dy ̧‚ Y2+ dy2) − ƒ(y1+ dy ̧, Y2)−ƒ(Y1, Y2+ dy2)+ƒ(Y1, Y2)• The right-hand members of these last two equations being identical, we must have also

This relation is true whatever be the magnitudes of dy, and dy,: if we proceed to the limit, by taking dy, and dy, less than any assignable magnitudes, and replace infinitesimal differences by differentials, we get

Expressing the theorem by partial differential coefficients instead of partial differentials, we have

or, as these partial differential coefficients are ordinarily expressed for the sake of brevity,

it is evident that the symbols d1, d2, of partial differentiation, may be permuted in every possible way: thus

dvdr,u = dv, dv, dvu = dy, dy, dyu = dy, dy, dyu = dv, vu,

or, in the language of partial differential coefficients,

COR. 2. The theorem which we have established in relation to partial differentiation of functions of two variables, may evidently be extended to the general case of a function of any number of variables: thus, in an expression

u being a function of y1, y2, Y3,... the symbols dy, dy, dyž‚. . . . may be permuted inter se in the same way as the symbols of quantity, A1, A2, A„,....in an algebraical product

Thus we see that the results are the same for both orders of

Successive Differentiation of an explicit Function of two
Functions of a single Variable.

49. Let u = f(y1, y2), y, and y, being each of them a function

Differentiating again, x being considered the independent variable, and observing that, for convenience of writing, we may put, being any expression functional of x,

But, since y,

=

+

dx dy, dx

D (du

dx dy

D (du

dy

dx

+

dx dy, dx

du D (dy,

(

dy, dx dx

D

dx dy dx dy, dx dx

(du). dy2+ du L (dy,

and y, are functions of x only, and not of any

du

+

dy, dx2

du d2y2

du

dy, dx2

Now is equivalent to a function of y, and y, only, not

dy,

involving the differential dy, thus, for instance, if u = y,y2,

2yy, where dy, does not appear. It follows there

fore that in the expression

D (du

dx dy

we may regard dy, constant without affecting results. Hence,

du

dy,

now occupying the place of u in (1), this formula gives