being equal; and similarly (d) dy" represents the mth dif

ferential of u, formed by making y to increase m times succes

sively by equal increments dy; and

dru dxm dyr

-m) dam dyr-m indicates the rth differential of u, formed by making a to increase m times successively by equal increments da, and y to increase (r-m) times successively by equal increments dy.

The principles on which this notation is constructed are so immediately applicable to functions of more than two variables, that it is unnecessary to do more than state the results as they occur in the following Articles.

Here however I must remark on a defect in the notation. If u = F(x, y), for instance, the same symbol d'u in the numerators of

d2u dx2

d2u

dx dy

(a),

dy2

means processes which, although of the same kind, may lead to results altogether different. In the first it means the second differential of u springing from two successive variations of x; in the second it represents d'u springing from a variation of x, taking place on the back of a previous variation of y; and in the third, d'u as originating in two successive variations of y. The brackets therefore are used in conjunction with the different denominators to indicate these variations of the same symbol, and are marks sufficiently distinctive to prevent confusion. But before we proceed further, we must prove the following proposition.

79.] If a function is differentiated many times in respect of independent variables which it contains, the result is the same, whatever is the order of the variables with respect to which it is differentiated: provided that it is differentiated the same number of times and with respect to the same variables.

For the sake of convenience, using symbols of differentiation,

d2u

dx dy) = (

d2u

dy dx).

dr (x, y, z,...) = F(x+dx, y, z, ...) — F (x, y, z, ...),

dr (x, y, z,...) = F(x, y + dy, z, ...) — F (X, Y, Z, ...) ;

(93)

(94)

therefore from (93),

dy dx F (x, y, z, ...) = F(x + dx, y + dy, z, ...) — F (x + dx, y, z, ...) − F(x, y+dy, z, ...) + F (X, Y, Z,...) ;

and from (94),

dx dyr (x, y, z,...) = F(x+dx, y+dy, z,...) — r (x, y+dy, z,...) -F(x+dx, y, z,...) + F(x, y, z,...);

and as the two results are identical, it is manifest that

dx dy r(x, y, z,...) = dy dx F(x, y, z,...) ;

or, writing the result according to the notation of Art. 46,

d2u

dx dy) dx dy =

d2u

dy dx) dy dx;

and dividing through by dy dx, we have

(95)

As the proof here given does not depend on the differentiations having been performed with respect to two variables only, it is plain that an analogous theorem is true for differentiations with respect to any number of variables; so that we may always interchange, in whatever manner it is convenient, the order in which the several differentiations are performed; as, for instance,

Hence also it follows, that if successive partial differentiations are performed on a function of many independent variables, by making x, y, and z to vary severally r times, s times, and t times, the order of these variations may be interchanged in any permutation, and the result is the same; thus if

Of this property, thus proved in the general case, some particular examples are subjoined.

80.] Differentiation of a function of two independent variables.

whence, taking the most general case, in which neither a nor

y is equicrescent, we have

the brackets indicating that the derived-functions within them

+3

(d) dy dry]

+ 3 [(d) dx d2x + (dd) (dx dy + d3y dx} + {
{
dx dy

+

dy2

dy3

(du) d3x + (du) d3y. (101)

Similarly may other total differentials be found; but the general term is too complicated to be of any practical use.

Now let the results be simplified by making x and y equicrescent, neither of which assumptions is inconsistent with the given equation; then

d2x = d3x = = 0; d2y = d3y

...

= = 0;

...

D2u =

dx2

dy3,

(102)

and so on; the law of the coefficients being the same as that of (1+)"; which may be proved to be true for positive integral values of the exponent, by a train of reasoning similar to that in Art. 55; whence the nth differential is

81.] By a similar process, if u is a function of many independent variables, x, y, z, ... all of which are equicrescent, we have the following series of equations:

If x, y, z, are not equicrescent, terms will have to be added analogous to those which, by reason of x and y being equicrescent, vanished from equations (100) and (101).