form xƒ(0), and the problem to be the elimination of a and its differentials between these two equations; we must first

day dx2,

replace d2x, d3x,

......

......

in terms of do, d20, d30,

by their complete values, and calculate da, and then we are at liberty to make any supposition that may be convenient as to y or being equicrescent. And a similar method, as is shewn in Ex. 3, 4, 5 below, must be adopted when two equations are given connecting x and y with two new variables.

and simplify the result by making the equicrescent variable. Or the question may be put thus,

To transform

lent, when

day x dy y

-

+

dx2 1-x2 dx 1-x2

= 0 into its equiva

(= cos-1x) is the equicrescent variable.

The above equation when complete is

But since is equicrescent, d200; whence, substituting, we

and express the result, (a) when neither y nor is equicrescent; (B) when is equicrescent; (y) when y is equicrescent.

The complete expression of the above equation is

(a)

(B)

(x)

y (d0)3 + dy (do)2 + 0 d2y do — 0 dy d20 = 0.

And if 0 is equicrescent, d20 = 0; whereby we have y (do)3 + dy (d0)2 + 0 d2y d0 = 0;

And if y is equicrescent, d2y = 0; whereby we have

Ex. 3. Eliminate x, y, dx and dy between

given x = r cos 0, and y = r sin 0; and state the result, (a) in the most general form; (3) when @ is equicrescent; (y) when r is equicrescent.

..

d2x = d2r cos 0 – 2 sin 0 dr do — r cos 0 (d0)2 — r sin 0 d20,

day = d2r sin 0 + 2 cos 0 dr do—r sin 0 (d0)2 + r cos 0 d20 ;

.. (dx)2+(dy)2 = (dr)2+r2 (d0)2,

d2x dy-d2y dx = r d2r d0—2 (dr)2 d0 — r2 (d0)3 — r dr d20.

Ex. 5. Transform into its equivalent, when z is equicrescent,

SECTION 5.-Successive differentiation of functions of two or more independent variables.

78.] Thus far we have considered the successive differentials of explicit functions of the form y = f(x). Let us now inquire into the form and the members of the several successive differentials of a function of many variables, all of which we will at first assume to be independent of each other: because we shall hereby treat of the most general case; and the particular one, when two or more of the variables are connected by an equation, will be included under it.

Let, as in Art. 49, the general form of the function be

u = F(X, Y, Z, .),

...

(90)

where x, y, z,... are independent of each other; and are there

fore such that a variation of one does not necessitate a variation of another. The symbols which we shall here adopt will be in principle the same as those which have heretofore been used in this Chapter; and they will be extended analogously to the peculiar circumstances of the present inquiry.

Now on referring to Article 46, it appears that a function of many independent variables is susceptible of variations of many kinds; thus only one of the variables may undergo a change; or two or more, or even all, may simultaneously change value; and consequently there may be either partial changes or a total change of the function. Hence have arisen partial and total variations. Similar changes take place in the successive variations of such a function, according as such changes are due to the variation of one, or more, or even all, of the variables; and, if the notation is adequate, it will indicate the particular kind of change which has taken place. To fix our thoughts, let us first take a function of two variables, and of the form

u = F(x, y).

(91)

Then, by Art. 46, if du represents the total change of u when x and y have both received infinitesimal increments dx and dy, and if (du) dx and (du) dy represent the changes of u when a

and y have each singly and separately received infinitesimal increments dr and dy, then

both x and y; and that they, as well as du, admit of being dif ferentiated again; and that of them there will be partial as well as total differentials. These several differentials must have distinctive symbols, and the distinction must be shewn in the symbols: if this object can be accomplished much confusion will be avoided.

Let the several successive and total differentials of Du be expressed by D2u, D3u,... Du,.... Then since the x- and y-partial

differentials of u have been represented by (du) da and

it follows that the

PRICE, VOL. I.

du

dy

S

dy,

the brackets being throughout employed to signify that the differentiations are partial. Similarly the

dy drz) dy, (dr; dx dy

dy) dx

and so on for the others. In accordance therefore with the

principles and notation of Art. 54, (d), (d) will severally

dy2

represent the second differential coefficients or derived functions of u, formed by separately making x and y to vary twice d2u

successively and by equal increments. And

dady) will represent the second derived-function of u, formed by making first y, and subsequently a, to vary by infinitesimal increments dy and dr. An analogous meaning belongs to symbols such as

y

(d), (d), ...; thus, for instance, by (

m

dyn

(dnu), (de2 ;) is meant

d4u

dx2 dy2

the fourth derived-function of u, which is calculated by making first to vary twice and by equal increments dy, and subsequently a to vary twice by equal increments dr. The order of factors in the denominator indicates the order in which the differentiations are performed: and in the next Article it will be proved that this order is indifferent. Hence also it follows that

(d"") da" represents the nth differential of u, formed by making

x to vary n times successively, the successive increments of a