4 
L FLUXIONS. 445 
~ Invers 16) ef — When the be ee of the sine and cosine are Iavena 
_ Method. . — __ sin." x cost » negative, we multiply the numerator by cos.2a Me 
—— S dz sin” 2008," 2 = Ta mon + sin? 2=1, and aia / Beart 1 
° m—1 , dx iy. dx dx 
+ aan t# ee Nt sin.” x cos.” 2 ~sin."=* xcos/at sin” 2008 8 
resolving the proposed fluxion into the two fac. By repeating this operation, we come to fractions hav- 
Gece tine eet nes aad ing in ing only sin. x, oF cos , in the denominator. P 
the same manner, we find m=N, as sin. cos. z= 4} sin. 2 x, the denomi- 
d c 1 
; ei wud 4S nator sin.” x cos.” x becomes gn Sin.” (22). i, 
* ‘ d x sin.” x Cos.” «= mtn 146. We shall conclude this branch of the subject, 
? + ‘ a 
; fhe” ; by fin the fluents of four of the more elementary 
+ aaah a sin.” x cos."— 2. circular ions. 
dz  dxsin.x 
J One of these formulz serves to depress the exponent The first is ——~= ———__; put = 2, and it 
- of the sine, Se, ae ae of tie caine and, by 2 ren T—cos.2z* Pu 00s. 7 = % an 
| their joint ication, the fluent may be found when a2. : : 
r Pye i cay Wo positive’ itkger “Hinatbert.” For becomes — 7—,- The fluent of this expression has 
: exam) 
ple, 
Sd x sins x c08.*.z = — + sin. xc08.3 x 
+3 faz sin. x 0s. x 
S dx sin. x 0032.2 = +sin# 2c0s,.24 fd 2 sin. 2, 
Observing, now, that fd x sin. = — cos.x, we find, 
ats enlceniogiall eo terpe./ die ts ern 
cos, 2 (— + sin.? 2 cos.? 24-2 sin.ex — 3 ms 
. 143, But if m and n are negative, these pede sab 
quire some modification. The first gives 
@).. 
desin.™ x sin.™—"x 
cos.” & mM—n COS" x 
m—1 dx sin." *z 
~ m— Tl cos.” x 
dz sin, x 
This brings the fluent to depend on that of 
cos.” x” 
or-of 2, according as mis an odd ot an even number. 
bi te a Sep aenb Cretan er tert tinea edhe 
comes ——;: the fluent of this is obvious ; the fuent 
Treslael aan olen aes 
~ The second mead gas le, by making 
n negative, and bringing the fluent in the second mem- 
ee tae ee allen, fh 
ly, changing n into 2— n, gives 
pdesint 2 sintt's — mmnt2 pdesin”™ 2 
cos? ¢ (n—1)cos"—2 r—tl cos,"—2 z¢° 
By this, the fluent sought is reduced to that of dz sin.” x, 
= to 228i." ; Re 
cong? according as n is even or odd. ‘The 
second is found by formula (I); the first is present- 
ly to be noticed. 
144. If we make m or n =0, we have 
. —cos, x sine | m—1 
S sunrdss = T af ax sins 
sin. 2cos."—"z n= 
S corvds= ~, + ft cose 
* 
dt _ —cos.2 m—2 da 
ain 2 (m—i)sin”—2 m—,) sna 
dx ” sin. 2 n—2 dx 
i 008," — (i—I)cos."—*z tn cos." x 
been given in Art. 119. Case I; thence 
SJ se (04a FEI = fy Coes) 
And as cos. «= 1 —2sin.2 4.2 = 2cos.2 $x —1(A- 
riTHMETIC Of Sines, formule (T) ; therefore, 1— cos. 
=2:sin# 4 2, and 1-+cos, x = 2 cos.* }.x, so that 
2. By a like process, putting cos, «=z, we find 
dx =i {wv 1+sin. x) 
cos.z U4/(1 —sin, x) 
1 — cos. 
If in the formula [pene a= tan.t} 2, we putie—x 
instead of x, (# being 180°), we get i 
1—sin. 
=tan,” —37)=—_.— > 
Ct terrae 
therefore, f° 22. =1. {tan (45 4 Fs) } 4 e!, 
3. Let the fluxion be “#008 In this case, the 
numerator is the fluxion of the denominator, therefore 
(Art. 114, Rule IL.) 
8 O88 of 2 =f decot.r=1. (¢ sin. 2). 
tan.z 
x 
sin, zy 
4. In like manner, 
S SS dx tan. x =f S>{S}- 
cos. # cot. x cos.x 
Of the Constant Correction of a Fluent. 
147. Let P be the variable part of the fluent of 2dx, o¢ the con. 
(z being a function of x), and ¢ the constant quantity stant correc. 
which ought to be added, in order that the fluent may tion of a 
be the most general possible, we have fz da =P+4c. fluent. 
While we regard the fluent merely as a function, of 
which the fluxion is to be identical with a 
fluxion, c may be any constant quantity whatever ; but 
when the fluent results from the solution of a particu- 
lar problem, it generally happens, that the constant 
pie c has to satisfy some condition, which restricts 
it toa inate value. If, for example, it be propo« 
