FLUXIONS

459

measures, and therefore may be used instead of them. When thefluxion of a quantity is itself variable its fluxion is called the se-cond fluxion of the first quantity, and is denoted by two dots overit; if the Second fluxion be variable, its fluxion is the third fluxion

and is denoted by three dots over it, &c. x,x,x, x, &c. representthe first, second, third, fourth, &c. fluxion of x. The higherorders of fluxions may be illustrated by fig. 4, 5, and 6; where if*■ denote AP and y, PQ, and if PQ flow along AP uniformly, theflukion of the absciss AP or x is constant, or always the same, or*, =a o, in all the figures, if AQ be a right line, as in fig. 4, theny =zrq, is also constant, for the angle q Qr = the angle A beingconstant, and the angle Q.rq, a right angle, also Q r, constant, r qis always the same, or/ = 0. But in iig. 5, r q or/ continuallyincreases; and in fig. 6, decreases more and more, hence in bothcases y has a second fluxion, which is positive in tig. 5, and nega-tive in fig. 6.

To Find the Fluxions of Flowing Quantities, calledThe Direct Method.

(17.) If a variable quantity -v be multiplied by a constant quan-tity a, the fluxion will be a x. For if a: increase uniformly, axwill increase uniformly, and a times as fast, hence the fluxion willbe ax, and thus in all cases the constant multiplier must be re-tained.

(IS.) The fluxion is not altered when a constant quantity isconnected to the variable one by the sign plus or minus; thus thefluxion of x -J- a is x . For since a is constant and only to beadded or subtracted, it does not affect the increase or decreaseof x.

(19.) The fluxion of the product of two variable quantities, asx y, is the sum of the products of each into the fluxion of the other,®**j+y*. Let AP, fig. 7, = x, and PQ — y ; then BP =x y = the flowing or variable rectangle, and let AD so move onAC, and AC on AD, that the intersection Q may describe thecurve AQ, and let Pp represent the fluxion of AP, and B b of AB°r PQ, then xy = the fluxion of the trilinear space ABQ , and p.vmat of the space AQP, therefore Xy -\-y x = the fluxion of thewhole rectangle PB, by art. 14. Otherwise. I.et the factors oftne product x and y, by flowing become x -|- x', and y -f- y', thenthe new or contemporaneous value of the flawing product will byOrultiplicatian be found equal to xy -j-x if -j- yx' x' y '; and sub-tracting the original product xy, the remainder xy' -f -y x' -\- x' >/wdl be the increment of the product generated in the same time as■q,? r y : but this increment is not uniformly' generated, for the part>t denoted by x'y' arises from an increase added to that by whichthe product is uniformly augmented, as may be thus shewn. Lets °me part as half of the time be taken, then the quantities are;/

* i-and y -|-, and, as above, the increment of the product

2 2

• xy' yx' x'y'

n the same time will be found-1--|—1-; in 4 of the time

i 2 4

y x ' x'y' xy’ yx' x'y'

I ls -1-1-, in | it is-1-[- See. Hence it is evi-

, 4 4 1 6 8 S G4

e nt that the part of the increment x y'-\-y x'-\-x'y' has arisen from

II uniform increase, and the part x' y' has wholly proceeded fromn augmentation to that increase, for it continually rises from 0® X V jn a duplicate ratio ; hence had there not been an accele-u et i increase, the quantity x'y' would not have been generated ;

'before if we write tiie fluxions/ and x instead of their measuresJ x', we have xy y x for the fluxion of x'y.diw^’) n, cans of the above, the fluxions df the continued pro-jy ct °f any number of flowing quantities may be found as of xyz,

_ x y z > &c. Put v — xyz, and %v = x y, then v—wz, and v

k z ; and i,, =xy -} - yx (art. 19.); and by substitutinga ; v plues of w and w, we have „) = xy a s xy -f-i y x, and inunilar manner the fluxion of tv x y z will be found equal to

h * !Bl i + xwy x -j -zyx-iu ; and thus for any mlm-r °« factors.

' *•) T he fluxion of a fraction — is easily determined from that

y.

of a product. Put — =:r, then x = xy, and * = xy yi

y

x xy

and y it = x — Xy, and v — — — —, and by substituting the

y y

x xy y x — xy x

value of x>, we have v = -= —-, the fluxion Of — .

y . r

y

(22,) From art. 19, and 20, the fluxion of the powers of flowingquantities are also determined, for if x and y (art. 19,) be equal,then the product x y becomes a - 2 and the fluxion xy -)- y x be-comes xx -|-x* — 2 x x, and thus the fluxion of xyz (art. 20,)is equal to 3 x*x when each of the factors is equal to x. And ill

general the fluxion of x w by the same, article is evidently equal to

- ~ |

x x repeated as often as there are units in n, that is equal to

n — 1 .n x x.

v (23.) Hence the fluxion is found when the index is a fraction ni

of x m ,which put equal?/, then y =

. n — 1

n — 1 . , . nx i

= n x »•, and/ = _

of y)

n — i

H -

my

Hi" — 1

m— 1

:x”, and in fluxions my* 1 b= (by substituting the value

(24.) In like manner is the fluxion obtained when the index is

negative, as of x~ 71 == —, which put equal y, then is yx 11 1,

n

x

71 7 % m J

and in fluxions x j -\- y nx

ixO, and/ = —

y nx 1 *

yn x

(by substituting the value of y )

x

— ft Xr «+1

a x — n — 1 .

X.

(25.) The fluxion of the hyperbolic logarithm of any quantity x

x

is equal to—, that is the fluxion of the quantity divided by thex

quantity itself. Let L be the principal vertex of an hyperbolaLV, Plate LXX1I. fig. 8, OR and OS the asymptotes, and OA= AL = 1, the parameter, and PM an ordinate parallel to AL.Then from the nature of the hyperbola and of logarithms, the spaceALMP is known to be the hyperbolic logarithm of OP ; put OP= .t, and PM = y, then the fluxion of the space ALMP is equaly -j_ the fluxion of AP, orjof x — 1 = y x, (art. 14, and 18,) =the fluxion of the logarithm of x: but from the nature of the curve

1

OA + AL = OP + PM ; hence 1 = xy, and y = —, therefore

x

X

y x — — = the fluxion of the hyperbolic logarithm of X.a?

(26.) Let m be the modulus of any other system of logarithms,and let e be the hyperbolic logarithm of any quantity x, then it iswell known that m z is the logarithm of x in the proposed system ;

but the fluxion of m z is equal to m £>, (art. 17,) and ss — —, (art.

x

m x

25,) hence m% — -, hence the fluxion of the logarithm of anv

.T

quantity in any system of logarithms is equal to the fluxion of thatquantity multiplied by the modulus of the system, and divided bythe quantity itself.

(27.) By means of logarithms the fluxions of exponential quan-tities