volved under the functional symbol, and that therefore any one may vary without necessitating a variation of the others, and that a variation of any one will cause a variation in the tree's growth, it follows that there may be three several variations of the effect, owing to the separate and several variations of the three producing causes. Thus, suppose the fertility of the soil to be increased, while the other two causes are unchanged, the tree's growth may be thereby increased; and similarly may it be increased by a change in either of the other two causes. Now although these three acting causes are so combined by means of the connecting equation, that when a finite change of one has taken place, a subsequent change of another may not have the same absolute effect on the tree's growth as if it had changed without the first having previously varied, yet such will not be the case when the changes are infinitesimal; because the infinitesimal variation of one acting subsequently to and on the back of the infinitesimal variation of the other, infinitesimals of a higher order, or products of infinitesimals, will be introduced; and these must be neglected by virtue of Theorem VI, Art. 9; and thus the absolute infinitesimal change in the tree's growth, due to the infinitesimal variation of any one of the three acting causes, will be the same, whether the other two causes have changed or not. And therefore, when all three have changed, the resulting change of the effect will be the sum of the three effects due to each of the three producing causes; of this however a mathematical and rigorous proof will be given subsequently. Thus we have variations of the growth of four distinct kinds: a variation due to each of the three causes acting separately, and the whole variation which is the sum of these three. This latter is called a total variation or a total differential, and the former are called partial variations or partial differentials; each of which, it is plain, is to be calculated on the supposition, that one variable only changes and that the others do not change.

Or again: consider a plane rectangle OPRS, see fig. 9, of which one side op = x, and the other os = y; and let u represent its area. Therefore,

u = xy.

Now the area may vary owing to a change in the length of either side, or to changes of the lengths of both. Suppose x to be increased by PQ = Ax, see Article 19; then the area is increased by the rectangle RPQw yax; which, when a be

=

comes dx, is y dx. Or suppose y to receive a finite increment, that is, os to be increased by STAy; then the increase of the area is the rectangle TR= xay; which, when the increase is infinitesimal, is a dy; hence the partial changes or increments of the rectangle are y da and x dy. But suppose both sides to have received infinitesimal variations; that is, a to become x+dx, and y to become y+dy, then the rectangle becomes

(x+dx) (y+dy) = xy + y dx + x dy + dx dy;

that is, the increment of the rectangle is y dx + x dy + dx dy, which terms severally represent the rectangles RQ, TR, UW; and therefore, when the increments are infinitesimal, the first two are rectangles of finite length but of infinitesimal breadth, and the latter, being a rectangle with infinitesimal sides, is but a point, which must therefore be neglected; and we have the total differential of u

= x dy + y dx

the sum of the partial differentials.

Bearing in mind what was said in Art. 19, and that Greek letters are used to signify finite changes or differences, and English letters infinitesimal variations or differentials, we shall find the following symbolization convenient.

Let u = r (x, y, z, ...) be a function of many variables, and let Du or Dr represent the total differential of u due to the variations of all the variables, and let

dru, or dyr express the partial change of u due to a change of x,

represent the ratios to the increments of the variables, of the several variations of the function due to the variation of the variables separately; that is, let them represent partial differential coefficients, or partial derived functions, the brackets indicating that they do so, and the variable in the denominator of the fractions being that, due to the change of which, the partial variation of u is calculated.

Thus, although the numerators involve the same symbol du, yet the same thing is not represented by it; for the du arising

from the growth of x into x+dx, may be a wholly different function from the du which comes from the growth of y into y+dy. If then we have occasion to use these symbols together, a mark of distinction is necessary, such as we have in du, du, ...; but when the ratios of the changes are required, the denominator in addition to the bracket is sufficient to indicate the origin of the function.

That we may have distinctive names for the differentials which arise in processes so different, I shall call them the x-, the y-, ... differentials, according as they arise from the infinitesimal variation of x, of y, ...; and I shall apply similar names to the several partial derived functions or differential coefficients.

Thus much as to the symbols, their nomenclature, and their meanings. Let us now consider the most simple case of a function of two independent variables.

47.] Differentiation of a function of two independent variables; and of the form u = F (x, y).

Let x and y receive the increments ar and ▲y; and let the corresponding increment of u be su; so that

Au F(x+▲x, y + ▲y) — F(x, y);

.. AU = F(x + sx, y + sy) − F(X, Y +▲Y) + F(X, Y+AY) - F(x, y);

let the variations of x and y be infinitesimal, then au becomes Du, which is the total change in u; and r(x+ax, y + sy) - F(x, y + sy) becomes du; because whatever difference there is between the first and second of these quantities, it is solely due to the change of x, y + sy being the value of y in both of them; and for a similar reason, F(x, y + ▲y) − F(x, y) becomes du; and therefore

that is, the total differential of a function of two independent variables is equal to the sum of the partial differentials.

in which formula dx and dy are the infinitesimal increments of

PRICE, VOL. I.

L

x and y.

Hence we have the ratio of the total change of the

function to the change of either variable: thus

Instead however of formally going through the processes of partial differentiation, and then of combining them, as I have above, I may take the result as exhibited in (40) or (41), and calculate the total differential immediately. Thus let

u = ay2+ bxy+cx2+ey + gx+k,

Du = 2 ay dy + b (x dy + y dx)+2cx dx + e dy+gdx;

and thus for other examples.

48.] Differentiation of an implicit function of two independent variables.

The principles applied in the last and preceding Articles enable us to differentiate an implicit function of two variables, and thereby to determine the ratio of the corresponding differentials of the variables, without the expression being put in the form of an explicit function of one variable *.

* To the above method of differentiating implicit functions it may be objected, that if

u = f(x, y) = c,

If

there can be no change in a due to a partial change in x, because u is constant; and for a similar reason there can be no variation in u due to a variation of y only; and therefore that x and y cannot vary separately, but must vary simultaneously; that one cannot change without the other, and therefore that there can be no partial variations of the function. To this it is replied by asking, What do we mean by a constant such as c on the righthand side of the equation? We mean that whose total variation is zero. therefore we consider c to be the sum of two quantities c, and c,, such that when a varies c, varies, and c, varies when y varies, but that c, and c, are so related that de1 + dc, = 0 = dc; then the total variation of f(x, y), being the sum of the partial variations, will be equal to 0, and the condition of f(x, y) being equal to c will be satisfied. Whereas then each partial variation taken separately is a violation of the condition expressed by the equation, yet the relation of the two when added together is such, as to be in accordance with the equation. The partial variation of the second neutralises the inconsistency of the partial variation of the first with the equation. This may be geometrically explained as follows. Suppose x2 y2 = + a2 b2

C

f(x, y)

= C,

which equation represents an ellipse the size of which depends on the value of the constant c, as well as on a and b. Suppose we consider y alone to vary, while a is constant; the point corresponding to x, y + dy, is no longer