FLt)
tlie Fluxion of A was reprefented by a, the Fluxion of A" would
be reprefented by n it X A — o , which is lefs than
w a X A — a (becaufe we fuppofe u to be lefs than o) and
therefore lefs than A" — a — u . But this is repugnant to what has been demonftrated. Therefore the Fluxion of A be- ing fuppofed equal to a, the Fluxion of A" rfiufl be equal to n a A"—'.
- tbe Fluxion of A being fuppcfcd equal to &, the Fluxion of ' A n will
u m - x Jr~\
n
Firft, let the exponent— be any pofitive fraction whatfoever, n
fuppofe A" =K; confequently A'»=K»; and the Fluxion of K being fuppofed equal to k, mahp 1 — ' = ntK« — ',
m
and k or the Fluxion of A" will be equal to
«K"
ma¥L
— __a a A
When — is negative, let it be
equal to — r ; and fuppofe A — r = K, or i = A r K, then taking the fluxions, rA'— ' a K -f- k A r = o, and k —
rh'-'aK = _ rA Ifl _.* x sA =-,
A' n
Suppofilg P to be the producl of any factors A, B, C, D, E, &c. and the Fluxions of P, A, B, C, &c. refpeclively equal to p t a>
b, c, fee. then will | =z~ + - + ~ + ^ , &c.
Xet Q_be equal to the product of ail the Factors of P, the firft A excepted, that is, fuppofe P^AQ; Suppofe R equal to the produdt of all the fadtors, the firft two, A and B, except- ed ; that is, let P = A B R, or Q_^. B R. In the fame man- ner let R r^. C S, Sz:DT, and fo on. Then the fluxions of Q> R> S, T, &V. being fuppofed refpectively equal to
q, r, s, t, &c. it follows that -^ := — 4- -£-
1 h
(becaufe- = g- +
0.'
B + R = ( becaufe R
D +
R/ A T
TV i "*" S""^ fJ "*" 5 "^~ T' and fo ™- Theref °re |
is equal to the fum of the quotients, when the fluxion of each factor is divided by the factor itfelf.
If the fat£tors be fuppofed equal to each other, and their num- ber be equal to n, then P = A", and by the laft propofition h na r , nPa
ij = — ; confequently p = ~j- = na A"— as was be- fore demonftrated.
4BC, &c. , , „ . _, „
and the bluxions of the refpeclive quantities
If P =.
KLM,&K.
be expreffed by the frnall letters p, a, I, c, &c. as before, then p a , b , e k I
A + B ^ C K . L'
At
,, ice
B+ C K L
For P K L M, &c. =3 A B C, fcrV. and t + - 4- - 4- -, &c. = — 4- =■ 4- g,SV. whence by tranfpofition |- — - 4. C5V. Maclaurin, ibid.
The notation we have hitherto ufed is the fame as Sir Ifaac Newton's in the 2d lemma of the 2d book of his principles. But it is generally more convenient todiftinguifh Fluxions from other algebraic expreffions, and in fuch manner, that the fe- cond and higher Fluxions may be reprefented fo as to preferve the original fluent in view. Hence Sir Ifaac in his laft method, reprefented variable or flowing quantities by the final letters of the alphabet as x, y, z, their firft, fecond, &c. Fluxions refpeitively by x, y, z, and x, y, z, &c. as is mentioned in the Cyclopaedia, under the head Fluxion, where the rules of the algorithm are alfo delivered ; but as this doflrine has been contefted and reprefented by the author of the analyft, as in- conceivable and fnphiftical, we thought it proper m.ve fully to explain and demonftrate the principles thereof, from Mr. Mac- laurin's excellent treatife on this fubieft. It is to be obferved, that the Fluxions of powers are commonly delivered in an algebraic form ; but this is not neceffary. The fame may be done geometrically by fuppofmg a feries of lines in geometric progreffion, the firft term of which is invariable. Then
FLU
if the fecond term be fuppofed to increafe uniformly ail the iubfequent terms will increafe with an accelerated morion. I he velocities of the points that defcribe thofe lines being com- pared, it may be demonftrated from common geometry, that
the Fluxions of
any two terms, are in a ratio compounded of
the ratio of thofe terms, and of the ratio of the numbers that exprefs how many terms precede them refpectively, in the progreffion. Thus' if A, B, C, D, E, kc. reprefent any lines in geometric progreffion, the firft term (A) of which is inva- riable, then will the Fluxion, for inftance, of E be to the Fluxion of D as 4 E is to 3 D, and the Fluxion of E will be to the Fluxion of B, as 4 E to B. The analogy between pow- ers in algebra and lines in geometric progreffion, is fufficiently known. Thus A being invariable may be called unity or 1 ; let B = a-, then will G=f**, D = *J, E — x\ &c. and confequently, the Fluxion of E or **•, will be to the Fluxion of B or x, as 4 x* is to a-, or as 4 «!, is to 1. Therefore if the Fluxion of x be expreffed by *, the Fluxion of #* will be ex- preffed by 4xix, agreeably to the common algebraic method of expreffion. Vid. Mr. Maclaurin's Flux. B. 1. Co, 6. See alfo the Prefent State of the Rep. of Let. Oct. 1735, pag. 248^ 249, &c.
If the Fluxion of B or the fecond term of the progreffion be invariable, every term of the progreffion will have Fluxions of as many degrees, as there are terms that precede it in the progreffion. And the increment of any term generated in a given time may be refolved into as many parts, as it has Fluxi- ons of different orders ; and each part may be conceived to be generated, m confequence of its refpective Fluxion. Hence Fluxions of all orders may be illuftrated and meafured. See Mr. MaclaurhhTtrnx., of Flux. B. I. c. 5, and 6. As to the higher orders of Fluxions, it is to be obferved, that when a motion is accelerated or xetarded continually, the ve- locity may itfelf be confidered as a variable or flowing quanti- ty, and may be reprefented by a line that increafcs or decrea- fes continually. When a velocity increafes uniformly fo as to acquire equal increments in equal times, its Fluxion is meafured by the increment which is generated in any given time. In this cafe, the velocity is reprefented by a line that is defcribed by an uniform motion ; and its Fluxions, by the conftant ve- locity of the point that defcribes the line, or by the fpace which this point defcribes in a given time. When a velocity is not accelerated uniformly, but acquires increments in equal times, that continually increafe or decreafe, then its Fluxion at any term of the time is not meafured by the increment, which it actually acquires, but by that which it would have acquired, if its acceleration had been continued uniformly, from that term for a given time. And in the fame manner, when a motion is retarded continually, the quantity by which it would be diminifhed in a given time, if its retardation was continued uniformly from any term, meafures its Fluxion at that term. While the point M defcribes the line E e, let the point Q_de- fcribe the line I ;', fo that I Q_ may be always equal to the fpace that would be defcribed by the motion of M, if it was con- tinued uniformly for a given time. Then IQ_fhall always reprefent the velocity of M s and the velocity of the
M
<o_
point Q mail reprefent the Fluxion of the velocity of M; which therefore is meafured, at any term of the time, by the fpace which would be defcribed by Q, with its motion at that term continued uniformly for a given time. The velocity of M is the Fluxion of E M; and therefore the velocity of Q reprefents the Fluxion of the Fluxion of E M. Thus, when a Fluxion of a quantity is variable, it may be confidered itfelf as a fluent, and may have its Fluxion, which is called the fecond Fluxion of that quantity. This may alfo have its Fluxion, which is called the third Fluxion of the firft fluent : and motions may be eafily conceived to vary in fuch a manner, as to give ground for admitting fecond Fluxions, and thofe of any higher order. And as the firft Fluxion of a variable quantity, at any term of the time, is meafured by the increment, or decre- ment, which would be produced, if the generating motion was continued uniformly from that term, for a given time; fo its fecond Fluxion may be meafured by twice the difference betwixt this increment, or decrement, and that which would be produced, if the acceleration, or retardation of the gene- rating motion was continued uniformly from that term for the fame time. Maclaurin, lib, cit. Seel. 70. Ibid. Se£t, 75. in fin.
See a farther illuftrarion of fecond and third Fluxions, in the fame author, (Chap. 3. and 4.) deduced from the conlidera- tion of the Fluxions of folids.
The author of the analyft has reprefented the notions of fe- cond and third Fluxions, as inextricable myfteries ; and in- deed, when fome fpeak of the velocities of velocities, &c. it is not eafy to fay what they mean. But it is to be obferved, that the firft Fluxion of any fluent, is not the velocity of that fluent, but the velocity of the motion bv which the fluent
