f LU
fe conceived to be generated. And in like manner, the fecond Fluxion of that fluent is not the velocity of the velocity of this fluent, but the velocity of the motion by which the quan- tity is generated that always reprefents its firft Fluxion, and fo of the reft. See Madmtrin, Lib. cit. Sect. 164. When the Fluxion of a quantity is variable it may be confider- ed as a fluent; and its Fluxion which is the fecond Fluxion of the propofed quantity may be determined by the preceeding propofitions. Thus the Fluxion of A being fuppofed equal to a the Fluxion of A A is 2 A a ; and if A be fuppofed to increafe at an uniform rate, or its Fluxion a to be invariable, 2 A a will increafe by equal fucceffive differences; confequently its Fluxi- on, or the fecond Fluxion of A A, will be equal to any of thefe differences, as to %a X A-f a — 2 A a-> or 2 a a. If be variable, let its Fluxion be equal to z, and the Fluxion of 2 A a (or fecond Fluxion of A A) will be 2 a a -f 2 A z. In the fame manner the Fluxion of A being conftant, the Fluxion of » A t-r* a. A"— i a, (.
or the fecond Fluxion of A" is n a x n — 1 x
1 ; the Fluxion of this, or
or n X « — 1 X a a A" third Fluxion of A n , is bXb - 1 X n — 2 x. a 3A*_ s. And the Fluxion of A" o f any order denoted by m, is n x n — 1 x n — 2 X n— .3, Eff c. Xa 1 " A " — m , where the fac- tors in the coefficient are to be continued till their number be equal to m. When « is any integer pofitive number, the Fluxion A", of the order n, is invariable and equal to n X n — 1 X n — 2 X n — 3, &c. X a". The quantities that reprefent thofe Fluxions of A" depend on a, which reprefents the Fluxion of A. When A remains of the fame value, the firft Fluxion of A" is greater or lefs in the fame proportion, as a is fuppofed to be greater or lefs ; the fecond Fluxion of A", is in the duplicate ratio of a ; and its Fluxion of the order m> is as a m . If a be variable, but z the Fluxion of #, or the fecond Fluxion of A, be conftant, then the fourth Fluxion of AA will be conftant and equal to 6 z z ; for we found, that the fecond Fluxion of A A was 2 a a + 2 A z ; the Fluxion of which is 4. (7 z -f- 2 a z, or 6 a z ; and the Fluxion of this is 6 z z. In like manner the fixth Fluxion of A> will be conftant in this cafe and equal to 9 z 1 .
The fecond differences of any quantity B are the fucceflive differences of its firft differences ; and as the Fluxion of B in- creafes when its fucceflive differences increafe; fo its fecond Fluxion, or its Fluxions of any higher order increafe when its fecond or higher differences increafe. If we arrive at differen- ces of any order that are conftant, the Fluxion of the fame or- der is conftant and is expreffed by that difference. Thus when A is fuppofed to increafe by conftant differences equal to a, and its Fluxion is fuppofed equal to a, the fecond difference of AA (or A-4-tf* ■ — 2 AA 4- A — a*)is2aa ) which is like- wife its fecond Fluxion ; and the third difference of A' is 6 a'^ which is its third Fluxion. When n is any integer and pofitive number, the Fluxion of A„ of the order n is equal to the Fluxion of any of its firft: differences of the order n — -2, and fo on. For the Fluxion of A ~\~ a — A" (one of the firft diffe rences of A ") of the o rder n — 1 is n x n — ix« — 2, &c.
XA+a"~ "~^ ' — Ax c" _I =bX»-i y.n — 2, &c. X a" ■where the coefficients are fuppofed to be con tinned till their number be n — 1, fo that the laft muft be 2. And this we found to be the Fluxion of A" of the order w, in the preceed- ing paragraph. In the fame manner the Fluxion of A -|- a ' — 2 A" -f- A — a (the fecond difference of A") of the order n — 2, is equal to the Fluxion of A-f- a —A of the order n — , 1 ; and confequently equal to the Fluxion of A" of the or- der n. Thefe Fluxions are invariable and equal to the laft or invariable differences. But in other cafes the Fluxions of A" of any order are lefs than its fubfequent differences of the fame order, but greater than the preceeding differences, as before mentioned. Madam-iris Flux. art. 720, feq. By fuppofing one of the variable quantities to flow uniformly it will have no fecond or higher Fluxions., and the higher Fluxi- ons depending on it will be expreffed in a more fimple man- ner. Thus the Fluxion of x being fuppofed conftant, the firft Fluxion of x n being n x n — 1 x, its fecond Fluxion will be n xn — i X x * x"— z y and its Fluxion of any order m will be n x n — 1 X n — 2 x n — 3, &c. X x m x" — w , where the factors in the coefficient are to be continued till their number be equal to in.
The fecond or higher Fluxions of quantities may be found by particular theorems (without computing thofe of the preceed- ing orders) as may be feen by the laft example. See farther in Madauriris Fluxions, Art. 734. Jnverfe method of Fluxions. By this method the fluent is found when the Fluxion is given ; and the rules are derived from thofe of the direct method ; as the rules in divifion and evolu- tion In algebra are deduced from thofe of multiplication and in- volution. As when a fluent confifts of a variable, and invari- able part, the latter does not appear in the Fluxion ; fo when any Fluxion is propofed, it is only the variable part of the flu- ent that can be derived from ii. If a- reprefent any Fluxion
FLU
that may be propofed, the variable part of the fluent will be equal to x : for fuppofing y to be any variable quantity, if
- +y could reprefent the fluent of i, then x -\- y would be
equal to x, and y —o, or y would be invariable, againft the fup- pofition. But fuppofing K, to reprefent any invariable quan- tity, than x •+■ K may generally reprefent the fluent of .i. If it be required to find fuch a fluent of * as (hall vanifh when
- is fuppofed to vanifh, it can be no other than a-; and if it
be required that the fluent fhould vanifh when x is equal to any given quantity a, then by fuppofing * -f- K to vanifh when x becomes equal to a j we fhall have a 4- K. = c, or K = — a; whence the fluent is * — a. In the fame man- ner, the fluent of — x may be generally reprefented by K —x. When a Fluxion that is propofed, coincides with any of thofe which were deduced from their fluents in the preceeding ar- ticles, the variable part of the fluent required, mutt coincide with that which was there propofed. As divifion in algebra leads us to frafiions, and evolution to furds ; fo the inverle me- thod of Fluxions leads us often to quantities, that are not known in common algebra, arid that cannot be expreffed by common algebraic fymbols. Madaurin's Flux. art. 735. We cannot here pretend to enter into a detail of the rules of the inverfe method of Fluxions. We fhall only obferve in ge- neral, that a Fluxion being propofed, its fluent may fometimes be found accurately in algebraic terms; but this is far from being always poffible; and recourfe muft fometimes be had to a converging feries. Thus, if nx—' x were propofed, the variable part of the fluent is found by adding unity to the ex- ponent of the power, dividing by the exponent thus increafed and by the Fluxion of the root. That is, the variable part of
the fluent of n x" — ' x will be
B-I-(-lX
a x
I = *". But
if the propofed Fluxion were — - , we cannot find its fluent
by this rule, but we may throw the Fluxion into an infinite feries by dividing a by a — x in the ufual method, and we fhall
find the quotient or " =: 1 4- .* 4- *-l 4. * 5 . ... a — x a a 1 T ,it o-c.
Hence fJL~i+ — + - a — x
i 1 x*x , T — — -f- &c. Now tire
fluent of each term of this feries may be found by the forego-
ing rule ; and therefore the fluent of
- will be expreff- which may
ed by the feries » + i-|. #_ 4- _?_ -f
2 a 3 u a 4 a > be of ufe for determining the fluent when x is very fmall in refpect of a; becaufe in that cafe, a few terms at the beginning of the feries will be nearly equal to the value of the whole But it often happens that the feries deduced in this method' converges fo flowly, as to be of little or no ufe. See Maclau- ,-in's Treat, of Flux. Art. 737, 744, 827. &;V%,de Summat. oener. p. 28.
Mathematicians therefore do not always immediatly recur to infinite feries, when it does not appear that a fluent can be affign- ed in a finite number of algebraic terms. The arches of a circle, and hyperbolic areas or logarithms, cannot be afflgned in alge- braic terms, but have been computed with great exactnefs'by feveral methods. By thefe with algebraic quantities, anv feg- ments of conic feaions, and the arches of a parabola are eafily meafured ; and when a fluent can be affigned by them, this is confidered as the fecond degree of refolution. When it does not appear that a fluent can be meafured by the areas of conic feaions, it may howeVer be meafured in fome cafes bv their arcs ; and this may be confidered as the third degree o'f refolution. If it does not appear that a fluent can be affigned by the arcs of any conic feaions (the circle included) it may however be of fome ufe to aflign the fluent by an area, or arc of fome other figure, that is eafily conftruaed or defcribed ; and it is often important, that the propofed Fluxion be reduced to a proper form, in order that the feries for the fluent may not be too complex, and that it may not converge at too flow a rate. See Maclaurin's Treatife of Fluxions, Book II. cb. 3. We may therefore conftitute three orders or claffes of fluents. Firft, fuch as can be accurately affigned in finite terms by com- mon algebraic expreflions. Secondly, thofe which can be re- duced to the areas of conic feaions, or to circular arcs and lo- garithms. Thirdly, fuch as can be affigned by hyperbolic or elliptic arcs. ^ The firft two claffes (confidering triangles and circles as conic fecrions) may therefore be meafured by the areas of conic feaions ; and the third clafs by their perimeters, or lines that bound them. Maclaurin, ibid. art. 798.
The fluent of
of the firft
that of
fluents of
y'f'4; X x, ^/XX^/I +
is of the fecond ; but the
=Si and
