{+(-)-(1-2 -2x)"}

Ay = sin-1

x+Ax

α

Ax

a9

a2

x2

a2

a3

a2

x2 dx

2

+

1

a3

by reason of Cor. II, Lemma II, Art. 22;

=

dx

x2

a2

{a2 — x2} + '

= sin-1

{a2 — x2}

and as this latter process is the shorter, we shall use it in the following similar problems, although the former is more direct and elementary and is equally applicable.

{a2 — x2} dy,

dx

{a2 — x2} } ;

(31)

In respect of the above differentials it is to be observed, that

-1

the sum of d. sin- and d.cos-1 is zero; which arises from

X
a

a

thing, and for the same reason, is true of d. tan

43.] Hence, if the radius is unity, we have

d.tan-1f(x) =

f'(x) dx
1 + (f(x))2 *

(39)

Similar results are true of the other functions.

44.] These results may also be obtained from geometry, by a process similar to that employed in Art. 39. See fig. 8.

▲y = a sin-1x = the arc PQ = RQ Sec PQR = RQ SCC PCA;

let us take the Napierian logarithms of both members of the

equation; and we have

log y sin 1x log x.

The number of examples which have been differentiated, including the types of all known algebraical, exponential, logarithmic, and circular forms, and all of which indicate the derived function to be a new function of x, is sufficient to justify the presumption of Art. 18, that the differential of f(x) is a product of two factors, of which one is da and the other is a finite function of x. It is in fact almost an inductive argument per simplicem enumerationem.

SECTION 2.-The differentiation of functions of many variables. 46.] On functions of many variables.

When many variables are involved in an equation, it is theoretically possible so to arrange them that one shall be a function of all the others, whereby it becomes an explicit function of many variables. And these latter variables may be independent of each other, or have such relations amongst them that a variation of one may involve a variation of one or more of the others, whereby the number of independent variables may be diminished; the former case, which is the more general, shall be first considered, and then such modifications shall be introduced into the result as shall adapt it to the latter, and as the mutual interdependence of the variables may require.

For the sake of illustration let us consider a tree, and let us suppose its growth to depend on three circumstances which are independent of each other, viz. the fertility of the soil, the rain that waters it, and the heat of the sun then, if the relation or law of connexion of these four things is expressed mathematically and in an explicit form, we shall have the growth of the tree a function of three variables; thus,

Growth of tree F (fertile soil, rain, solar heat).

And if the law is known which connects the single effect with the three producing causes, then F, the symbol of the form of the function, is known.

Now observing the independence of the three variables in