Pirect” 
Methal 
—Y—" asa function of sin w; and i 
406 
considered as a function of its tangent. Put x fur the 
and w for the arc. Then we may consider u 
sin. w and cos. ™ as 
functions of x. Now, by Rule (D), supposing « a 
function of sin w, we have 
dues ti: (1). 
cos. 
And since by the Aniruneric of Sines, sin, x= 2 cos. u, 
therefore idering sin. « as a function of z, we have, 
by rule 
d (sin. u)=d 2 cos, w—xd(cos.u) (2). 
But cos.*# +4 sin.*u= 1; , by rule (A), 
cos. u d(cos,u) + sin. wd (sin. ~) = 0, and d (cos, u) 
= — 5 4(cin. u)=— 2d (sin, u); hence, fromequa- 
tion(2) we have, 
d (sin. u) = d x cos. u—zx* d (sin. 2), 
and (142*) d(sin.u)=dzcos.u. (3. 
From equations (1) and (3), we have, by rule (I) 
z 
1+? 
And since 1 4 2* = sec.* x = 
du =cos,*udz. 
Ex. 2. If we suppose u to be such a function of « that 
sec. w =, then, cos, w=. Now regarding u as a 
funetion of cos, u, we have, by rule (D) du =e), 
dix 
, we have also 
cos. *u 
dx +5 dz < cos.udx 
#'sin.u ~ sec.*« sin.u ~ sec. usin. u’ 
Hence putting tan. u for ~~, we have also 
Rie dz dz 
= sec. u tan. uw r4/(2—1) 
Ex. 3, Let cot. u=2 ; thentan, pay Tr Now we have 
x 
found that dw = d (tan. «) cos.* u, (Baamplp 1.) but 
d (tan. v) =—S; therefore, du= — =< «or 
= Cos. % 
since z= ——, 
sin. u 
du= 
du 
dux=—d rsin.? ae! Ss ; 
1+. 
ons If cosec,u= 2; then, from the formula, 
i = 1 
du= HO), and sin. u=—, we easily find 
dx da 
G0 =a dackens u  2/@—l) 
Some Applications of the Theory. 
36. Before we 
the theory of fluxions, we shall.give a few examples of 
its application to the investigati of analytical and 
ical formule ; ém the principle de- 
monstrated in Art. 28, namely, if two functions of 
variable ity be uni 
i acpaheh Bok laa 
FLUXIONS. 
By slenvenis Gereaiats i 
joan Fete tH. be i inink oF peal 
and therefore, 
a( =a) 
ia =t+ eat fat vee +2. (a) 
where n denotes the number of terms of the series. As 
the members of this equation are functions of the 
same variable quantity 2, their fl must be equal. 
cay H) fluxion of the first member is by rules (A) 
an 
(1— 2) fi—(n +1) et de42(1—2) de 
o>) 
| feat} dx 
1—z)* 
And. the fluxion of the second mémber is 
Dxp 2rd x4 3822 wove pne—idey - 
Hence, esting. Ore expressions equal to one another ; 
s 
also, diyidin by d x, and multiplying by x, we get 
x { 1—(n+ Dee tit 
= © 
(1—z)* ; 
242 22438 44at 2... 4n2% 
formula may be easily verified by multiplying 
the second member of the equation by the i 
of the first. - 
Let us now, in order to abridge, put 
xa +1)2"+ nar tl 
aa (i—2)* 
Then, by formula (4), 
Nasr42a' +3 34471... tnz, (c). 
then, taking the fluxion of each side as 
noting briefly d (X’ x X” dx, where X" is a func- 
tion of x, which may be found by rules (F), (H), and 
(1), we have 
X"dr=da4+2ardzr43*atdzr.., +n*2—"de; 
reer hence, dividing by dx, and multiplying by x, we 
X" eae estesg eat... 4 niat, 
From So by which formula (4) and (c) 
have been deduced from formula (a), it appears’ that if 
we put |—~ =X, and compute the series of functions 
Xamaprpaipaet ice. fe"; 
XX exq2z* 43254424... 042; 
Xa mar42? x? 4 $2734.47 af, . nites 
X"erar+23 24332344324... 439; 
&e, &e. 
where n may be any whole number whatever, and x 
any quantity greater or less than unity. In the parti 
cane Of don}, the forratile-g0ihol banana. 
plicable, because then the numerator and denominator 
of the expressions Xz, X’ w,&c. vanish at the same 
time. We shall shew, farther on, how quantities ha« 
Vig Oe Propety sre to Oe a 
hen z is a proper fraction, and nthe number of 
terms infinite, the funetion X becomes simply =~. Tn 
this case, 
and de« 
a 
—— ee 
