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Now, the rectangle consists of the two trilinear spaces APQ, ARQ, of which, the

fluxion of the former is PQ × pp, or yx,

that of the latter is

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RQ × Rr, or xy, by art. 8; therefore the sum of the two xy + xy, is the fluxion of the whole rectangle ry or ARQP.

The Same Otherwise.

16. Let the sides of the rectangle, x and y, by flowing, become x + x and y +ý: then the product of these two, or xy + xý + yx' +'ý will be the new or contemporaneous value of the flowing rectangle PR or xy: subtract the one value from the other, and the remainder, xý + yx' +, will be the increment generated in the same time as 'orý; of which the last term a'ý is nothing, or indefinitely small, in respect of the other two terms, because x' and are indefinitely small in respect of r and y; which term being therefore omitted, there remains rý + yx' for the value of the increment; and hence, by substituting * and for x' and ý, to which they are proportional, there arises xy+yx for the true value of the fluxion of xy; the same as before.

17. Hence may be easily derived the fluxion of the powers and products of any number of flowing or variable quantities whatever; as of xyz, or uxyz, or vuxyz, &c. And first, for the fluxion of xyz: put p= xy, and the whole given fluent xyz = q, or q = xyz = pz. "Then, taking the fluxions of y = pz, by the last article, they are q = pz + px; but P = xy, and so p = xy + xy by the same article; substituting therefore these values of p and p instead of them, in the value of q, this becomes q = xyz + xyz + xyż, the fluxion of xyz required; which is therefore equal to the sum of the products, arising from the fluxion of each letter, or quantity, multiplied by the product of the other two.

Again, to determine the fluxion of uxyz, the continual product of four variable quantities; put this product, namely uxyz, or qu=r, where q = xyz as above. Then, taking the fluxions by the last article, r = qu + qu; which, by substituting for q and q their values as above, becomes r = uxyz + uxyz + uxjz + uxyż, the fluxion of uxyz as required: consisting of the fluxion of each quantity, drawn into the products of the other three.

In

In the very same manner it is found, that the fluxion of vuxyz is vuxyz + vùxzz + vuxyz + vuxjz + vuxyż; and so on, for any number of quantities whatever; in which it is always found, that there are as many terms as there are variable quantities in the proposed fluent; and that these terins consist of the fluxion of each variable quantity, multiplied by the product of all the rest of the quantities.

18. Hence is easily derived the fluxion of any power of a variable quantity, as of x2, or x3, or x4, &c. For, in the product or rectangle xy, if x = y, then is xy xx or x2. and also its fluxion xy + xxx + xx or 2xx, the fluxion of x2.

Again, if all the three x, y, z be equal; then is the product of the three xyz=x3; and consequently its fluxion xyz+xjz+xyż= xxx+xxx+xxx or 3x2*, the fluxion of x3. In the same manner, it will appear that

4x3x, and

the fluxion of x+ is = 4x3x,

the fluxion of x5 is = 5x+x, and, in general,
the fluxion of x" is = nxx;

where n is any positive whole number whatever.

That is, the fluxion of any positive integral power, is equal to the fluxion of the root (*), multiplied by the exponent of the power (n), and by the power of the same root whose index is less by 1, (x-1).

And thus, the fluxion of a + cx being cx,

that of (a+cx ) is 2cx x (a + cx) or 2acx + 2c2xx, that of (a+cx2)2 is 4cxxx (a + cx2) or 4acxx + 4c2x3x, that of (x + y) is (4x+4yi) × (x2 + y2), that of (x + y2)3 is (3x+6cyỷ) × (x +cy2)2.

19. From the conclusions in the same article, we may also derive the fluxion of any fraction, or the quotient of one variable quantity divided by another, as of

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For, put the quotient or fraction =q; then, multiply

y

ing by the denominator, x = qy; and, taking the fluxions, * ≈ qy qỷ, x +gy, or qy = qy; and, by division,

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xj

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That is, the fluxion of any fraction, is equal to the fluxion of the numerator drawn into the denominator, minus the fluxion of the denominator drawn into the numerator, and the remainder divided by the square of the denominator. xy axy - axy So that the fluxion of

ax

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xy

is a x y

y

or

y2

20. Hence too is easily derived the fluxion of any negative integer power of a valuable quantity, as of ", or

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which

is the same thing. For here the numerator of the fraction is 1, whose fluxion is nothing; and therefore, by the last article, the fluxion of such a fraction, or negative power, is barely equal to minus the fluxion of the denominator, divided by the square of the said denominator. That is, the fluxion of x-",

-n

1

ΟΙ is

כ היל.

nxx
x211

or

nx

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nx *;

or the fluxion of any negative integer power of a variable quantity, as ", is equal to the fluxion of the root, multiplied by the exponent of the power, and by the next power less by 1; the same rule as for positive powers.

The same thing is otherwise obtained thus: Put the then is qan = 1;

proposed fraction, or quotient

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yx+qnxx=0; hence gr"-qnx"; divide by .", then

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21. Much in the same manner is obtained the fluxion of

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any fractional power of a fluent quantity, as of ", or

For, put the proposed quantity" =4; then, raising each side to the n power, gives x = q^ ;

taking the fluxions, gives manq1q; then

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=

mxm-1 mxm-x m

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=

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Which is still the same rule, as before, for finding the fluxion of any power of a fluent quantity, and which therefore is general, whether the exponent be positive or negative, integral or fractional. And hence the fluxion of art is ax*x;

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ar2

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22. Having now found out the fluxions of all the ordinary forms of algebraical quantities; it remains to determine those of logarithmic expressions; and also of exponential ones, that is, such powers as have their exponents variable or flowing quantities. And first, for the fluxion of Napier's, or the hyperbolic logarithin.

C

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23. Now, to determine this from the nature of the hyperbolic spaces. Let A be the principal vertex of an hyperbola, having its asymptotesCD, CP, with the ordinates DA, BA, PQ, &c, parallel to them. Then, from the nature of the hyperbola and of logarithms, it is known, that any space ABPQ is the log. of the ratio of CB to CP, to the modulus ABCD. Now, put 1 = CB or BA the side of the square or rhombus DB; m = the modulus, or CB X BA; or area of DB, or sine of the angle c to the radius I; also the absciss CP = ~, and the ordinate PQ = =y. Then, by the nature of the hyperbola, CP X PQ is always equal to DB, that is, rym; hence and the fluxion of the space, xy is = pagp the fluxion of the log. of x, to the modulus m. And, in the hyperbolic logarithms, the modulus m being 1, there

y =

ทาง

x

mx
X

fore,

fore is the fluxion of the hyp. log. of r; which is therefore equal to the fluxion of the quantity, divided by the quantity itself.

Hence the fluxion of the hyp, log.

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24. By means of the fluxions of logarithms, are usually determined those of exponential quantities, that is, quantities which have their exponent a flowing or variable letter. These exponentials are of two kinds, namely, when the root is a constant quantity, as ex, and when the root is variable as well as the exponent, as y`.

25. In the first case, put the exponential, whose fluxion is to be found, equal to a single variable quantity z, namely, z = e3; then take the logarithm of each, so shall log. z = xx log.e; take the fluxions of these, so shall =*xlog.c by the last article; hence = z x log. e = ex log. e, which is the fluxion of the proposed quantity ex or z; and which therefore is equal to the said given quantity drawn into the fluxion of the exponent, and into the log. of theroot. Hence also, the fluxion of (a + c)" is (a + c)"x x nx x log. (a + c).

26. In like manner, in the second case, put the given quantity yz; then the logarithms give log. z = x× log.y, and the fluxions give

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= log. y + xx ; hence

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y

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(by substituting p1 for z) yˇż ×

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log. y+xy-1, which is the fluxion of the proposed quantity y; and which therefore consists of two terms, of which

the

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