OF FLUXIONS AND FLUENTS.

ART. 49. In art 4', &c. is given a compendious ta ble of various forms of fluxions and fluents, the truth of which it may be proper here in the first place to prove.

50. As to most of those forms indeed, they will be easily proved, by only taking the fluxions of the forms of fluents, in the last column, by means of the rules before given in art. 30 of the direct method; by which they will be found to produce the corresponding fluxions in the 2 column of the table. Thus, the 1st and 2d forms of fluents will be proved by the 1st of the said rules for fluxion; the 3d and 4th forms, of fluents by the 4th rule for fluxions; the 5th and 6th forms, by the 3d rule of luxions; the 7th, 8th, 9th, 10th, 12th, 14th forms, by the 6th rule of fluxions: the 17th form, by the 7th rule of duxions: the 18th form, by the 8th rule of duxions. So that there remains only to prove the 11th, 13th, 15th, and 16th forms.

51. Now, as to the 16th form, that is proved by the 2d example in art. 98, where it appears that ✔ (dx-x2) is the fluxion of the circular segment, whose diameter is d, and versed sine x. And the remaining three forms, viz. the 11th, 13th, and 15th, will be proved by means of the rectifications of circular arcs, in art. 96.

rai

n

52. Thus, for the 11th form, it appears by that art. that the fluxion of the circular arc z, whose radius is r and tangent t, is z = then is tnx

Now putt = x2

12-1.

or t2 = x", and a =

x, and r2 + 12 = a + x", and ż=

-z, and the fluent is

x

; hence

2

or=

X arc to

n√a

2z 2

an na

radius 1 and tang.

in n°. XI.

53. And, for the latter form of the fluent in the same n°;

because the co-efficient of the former of these, viz.

double of the co-efficient of the latter, therefore the arc

nva

in the latter case, must be double the arc in the former. But the cosine of double of an arc, to radius 1 and tangent t,

is

54 Again, for the first case of the fluent in the 13th form. By art. 61, the fluxion of the circular arc z, to radius r

then these two being substituted in the value of ź, give

=

x;

(ax"); consequently the given fluxion

2√(a-x")

2

2

2

is

- , and therefore its fluent is

-z, that is

n

n

n

arc to sine

form XIII.

as in the table of forms, for the first case of

form, is the half of the co-efficient in the former case, there

n

fore the arc in the latter case must be double of the arc in the former. But, by trigonometry, the versed sine of double an arc, to sine y and radius 1 is 2y2; and, by the former case

21"

2y2 = ; therefore X arc to the versed sine

fluent, as in the 2d case of form XIII.

56. Again, for the first case of fluent in the By art 61, the fluxion of the circular arc z,

; hence s (s2 - 1) =

S1 (32 — 1)

2 r

or 33 =

va

a

(-a), and s

then these two being substituted in the value of ż, give ż or

case, therefore the arc in the latter case must be double the arc in the former. But, by trigonometry, the cosine of the

double arc to secant s and radius 1, is

32

2

1; and, by the

1

1:

n/a

2α- xn

to cosine

former case,

; therefore X arc

-is the fluent, as in the 2d case of form xv.

2

a

Or, the same fluent will be X arc to cosine✔ be

cause the cosine of an arc, is the reciprocal of its secant. 58. It has been just above remarked, that several of the tabular forms of fluents are easily shown to be true, by taking the fluxions of those forms, and finding they come out the same as the given fluxions. But they may also be determined in a more direct manner, by the transformation of the given fluxions to another form. Thus, omitting the first form, as too evident to need any explanation, the 2d form is ż=(a+ xn)-1-1x, where the exponent (n-1) of the unknown quantity without the vinculum, is 1 less than (n), that under the same. Here, putting y the compound quantity a+ then is y=n", and =

59. By the above example it appears that such form of fluxion admits of a fluent in finite terms, when the index (n-1) of the variable quantity (x) without the vinculum, is less by 1 than n, the index of the same quantity under the vinculum. But it will also be found, by a like process, that the same thing takes place in such forms as (a+xn)m xenix, where the

exponent

exponent (cn 1) without the vinculum, is 1 less than any multiple (c) of that (n) under the vinculum. And further, that the fluent, in each case, will consist of as many terms as are denoted by the integer number c: viz. of one term when c = 1, of two terms when c 2, of three terms when c = 3,

and so on

=

ya, and its fluxion

60. Thus, in the general form, ż= (a+x") mxn−1, putting as before, a+x"= y; then is x” = nx”—1x=y, or x2-1x= -1 Y and xen-1

n

(y-a)-y; also (a+x")m

1

ym: these values being now sub

Now, if the compound quantity (y-a-1 be expanded by the binomial theorem, and each term multiplied by yTM, that fluxion becomes

&c.); then the fluent of every term, being taken by art. 36. it is

y

n

--

&c.), putting d =

d-2

་

+c, for the general form of the fluent : where, c being a whole number, the multipliers c-1, c-2, c-3, &c. will become equal to nothing, after the first c terms, and therefore the series will then terminate, and exhibit the fluent in that number of terms; viz. there will be only the first term when c = 1, but the first two terms when c= 2, and the first three terms when c = 3, and so on.-Except however the cases in which is some negative number equal to or less than c: in which case the divisors, m+c, m+c-1, m+c - 2, &c. becoming equal to nothing, before the multipliers c1, c2, &c. the corresponding terms of the series, being divided by 0, will be infinite and then the fluent is said to fail, as in such case nothing can be determined from it. 61. Besides this form of the fluent, there are other methods of proceeding, by which other forms of fluents are derived, of the given fluxion ¿ = ́a + x”)mx, which are of use when the foregoing form fails, or runs into an infinite series ; sowe results of which are given both by Mr. Simpson and Mr. Landen. The two following processes are after the manner of the former author.

62. The given fluxion being (a+x")men; its fluent may

be

be assumed equal to (a + ")m+1 multiplied by a general series, in terms of the powers of a combined with assumed unknown co-efficients, which series may be either ascending or descending, that is, having the indices either increasing or decreasing.

viz. (a+xn)m+1 X (Ax+Bx"="+cx2¬ ̈+Dx2¬3+&c.) or (a+x)+1X (Ax+B++cx2+2 +Dx2+3 +&c.).

And first, for the former of these, take its fluxion in the usual way, which put equal to the given fluxion (a+xn)mxn−1, then divide the whole equation by the factors that may be common to all the terms; after which, by comparing the like indices and the co-efficients of the like terms, the values of the assumed indices and co-efficients will be determined, and consequently the whole fluent. Thus, the former assumed series in fluxions is,

mx

n (m + 1) xnx (a + x1)TM × (AX" + BX' + CX-2 &c.) + (a + xn)m. 1X (rax+ (r−3) B1+ (r-2s) cx3-2-1 &c.); this being put equal to the given fluxion (a+x)mxcn-1x, and the whole equation divided by (a+x")"x-1; there results n (m + 1) x′′ × (▲x2+BX~ + cx22 + Dx2-2 + &c.) } +(a + x2)×(rAx' +(r− s)BX'~+(r−2s)cx2¬2&c). Hence, by actually multiplying, and collecting the co-efficients of the like powers of x, there results

z(m+1) } Axr+n+n(m‍+1) Į Bx‍tn—s+n(m+1) { cxr+n=2&c.

+-rs

=0.

+r-255 +(r—s) αвxTM¬3 &c. Here, by comparing the greatest indices of x, in the first and second terms, it gives r+ n = cn, and r + n − s = r; which giver (c-1)n, and n=s.

=

Then these values be

ing substituted in the last series, it becomes
(c+m)naxen+(c+m−1)nBxn−n+(c+m−2)ncxen-2"&c.
-zen+(c-1)naszenn+(c2)naвx-2n

&c.

} = 0. Now, comparing the co-efficients of the like terms, and putting c+m=d, there results these equalities :

1

c-1.0A

c-1.a

d-1.dn'

c-2.ab

c-1.c-2.a2

d-2

=+d-1.d-2.dn'

&c.; which values of A, B, c, &c. with those of r and s, being now substituted in the first assumed fluent, it becomes

1 c-1.ac--1.c--2.a2 c-1.c-2.c-3.a3

dn 1 d—1.x" d-1.d-2.x3n d-1.d-2.d-3.x3 +&c. the true fluent of (a+xn) men-1, exactly agreeing with the first value of the 19th form in the table of fluents in my Dictionary. Which fluent therefore, when c is a whole positive number, will always terminate in that number of terms; subject to the same exception as in the former case. VOL. II. Thus,

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