_ 
444 
formula fudi=ut—fidu, and making s=1." (x), 
= and df=rd-z, so that t= fzds, and dw= 
w L=* (2) iy we have 
dx 
fries (= Ln(a) feda—nfms (=) feds, 
mynd hs ny ee og ap eatpuas A 
established, the proposed fluent is is form le 
to on another of the same kind, in which the 
exponent of the logarithm is an unit less. 
Thus, if=—2", we have 
freee @=ayi mary foo (2) .2" de. 
ing the fe Ja to this last fluent, and agai 
PR Sr ar tease > oe; we qt ‘we 
Seni (sz) dz= 
, fhe) nl?—(2) , x(n—l )Le*(@) : 
oe eet eri eee te 
187. But if » is a whole negative number, we must . 
| sp lorasbayer udi=ut—Jid u, so as to increase 
exponent of the logarithm, This will happen, if, 
in the expression 
z2 
[*(2) = —n+1 
d. > 
we make 2a = u, and L— (2): = = dt, by which 
I—*4+4(2) | 
—n+l 
x (—n4$1)1=*(2)%, 
=, for we then have 
zdz_ gn 
I) =~ —n +1 
— L-* +! (x).d (22). 
Let us suppose z = 2”; then this formula becomes 
xdz — gt! m+1 ~amdx 
Tz) ~(@—1) b=" (2) Fal P@’ 
By transforming this last fluent in the same manner, 
and again the fluent that thence results, and so.on, we 
wd 
at last make the proposed fluent depend on nO 
L741 (2) 
Now, put 2+? = +, then L. (2) = =), and a"dz= 
dz 
meV therefore, 
wdx dz edu = fra: 
Te) ie) =" ou 
pepe A agg The fluent of this last func- 
tion can only be expressed by a series, as we have al- 
ready observed in art. 134. 
198. When » is a positive or negative fraction, the 
fluent may be made to depend upon a similar fluent, in 
which n is between 0 and +1, or —1,. This last can 
only be expressed by an infinite series. 
Of Circular Functions. 
Of circular 
functions, 
139. 
of such expressions as coritain trigonometrical fune- 
There are several methods of finding the fluents - 
FLUXIONS. 
arc, its fluxion may be transformed into an alge. © 
braic function, . For example, let sin. «= 2, then 
dz 
cos. t = 4/(1—2"), dz=-—_—__; 
sip Va)" 
sin.” «. cos." 2dx=2"dz(1—2) a 
1. If n is an odd number, the radical in the trans« 
formed expression di i 
2. Ifm isan odd number, then the exponent of z out 
of the binomial, when increased by unity, will be a 
multi le of 8, it exponent ies, Mnausial Thus, one 
ie first) of art. 126, will be satisfied ; 
and therefore the fluxion may be made rational. 
8. If m and m are even numbers, then the second 
condition of art. 126. will be satisfied, As an example 
of this method, f‘sin.} x dx, 
dz ry 
VO—*) ~ 
RS. II. Method. It follows, from art. 26. rule D, 
— i con. «(8 ascot x) 4c 
_fizcos kas + sin. ke +6, 
\ 
[asrsin kes =F coke +0. yest 
Now, we have shewn 24 crip. « or Sines) how to 
develope the powers of sin. x and cos. « into series, the 
terms of which are multiples of 2; every 
fluxion of the form cos.”"2.dx, or sin.” xd.x, may be 
transformed into a series of futions, of the forms 
dx cos. kx, dx sin. kx; and hence the fluents may be 
found from the preceding formule. . Thus, because 
ae 5 pete 
Cos. & = ‘7 08. ee ee a ee 
(Anirumetic of Sines, Formule (S), therefore, 
nT Bis g: ‘ 
ross 2d 2 = 5 sin. 524 Gesin. 824 sin, & 
This method “is often used, because it is easier to find 
the sines and cosines of the multiples of an are than the 
powers of its sine and cosine. As-the expression 
cos.” x sin." z, may also be resolved into a series, of 
which the terms are the sines and cosines of the multi- 
ples ot Se the fluent of cos.” x sin.” xd x, may be found 
as in ere 
pc E on Pe ccbon Noe Sah Hotta 
may be as i i ‘or 
m of at. 195; bad tae the eaets of ony Sixxions 
into which enter, may be found by art. 132---134, 
142, IV. Method. This proceeds by the formula 
Sudt =ut—ftdu, which we have already so often 
4 ed, the fluxion be d w sin.” x cos." x, which 
oo dee gig ee protess oe 
sin."—! x; then, putting vu = sin.”— 2, t= 
dx sin. z con't a oon which it follows, that ¢ = 
ae and du = (m—1) dx cos. # sin." 2, we 
ve ; 
* sin” ze . 
UJ as sins cosa = = SS cost x 
| — _frosrtrasinm ad x, 
this 
lue cos." # (1—sin,?z), 
1 
for cos.n+2.2, its va= 
transposing, we find 
