the fluent of by the circle and- logarithms. 
ays 
John Bernoulli Taiowed in the same path as Leibnitz, 
but their methods were very laborious: The theory 
by the discovery of Cotes, 
i noticed, (art. 118.) 
On this branch of the subj see John Bernoulli, 
pm: ; Cotes, Harmonia Mensurarum; De Moivre, 
iscellanea Analytica ; Simpson, Epeeue oe several sub- 
jects, awd Fluxions, Vol. U1. ; Landen, Mathematical Lu- 
cubrations, Part VIT. Euler, Inst. Cal. Diff: et Integra- 
lis, 
Of the Fluents of Irrational Functions. 
122. We have shewn how the fluent of every rational 
algebraic function may be found ; the same method will 
apply to all such irrational functions as can be rendered 
rational by transformation. Let us consider, in the 
first place, fluxions, in which the terms are singly ra- 
dicals, such as : 
Soft fe xox x* 
t +42 ” 
it is easy to see that by making x = 2‘, the irrationa- 
oe 5 ma and as dz = 62° dz, the fluxion is trans- 
formed to 
6(2"4 + 2% 4+ 2) dz m 62zdz 
( wat =6z ek thal ager Ie 
which presents no difficulty. 
Let the fluxion be “= da; put x ==, then ds =: 
22zdz, and the fluxion is transformed to 
ads _o4 2Qdz 
Bal en ea 
of which the fluent is 2 z 4 log. (z—1) —log. (z++1), 
or 
c/t—c 
123. We are now to consider any function whatever 
affected by the radical 4/(A + Bax ==Cat); which 
may be also expressed thus, Voy (% + Ge) 
There will be two cases, according as #* is positive ‘or 
ve.« 
nse [When the i ' 
we radical has the form 4/(a4-b24-2*); 
V(@ 462+) =e -b2, or = sez, 
hence we find a +62 == 2x2 + 2', 
_ #—a 
* = bo2? 
2(b2>= a= 2) 
lien (roa 
® we tp that the factors are 
4a+4b2— 
be positive, and ¥(6* 4. 44) a real quantity. 
FLUXIONS. 
Thus the radical, or r=, will be rendered ra- Ini 
tional, ae swollen he peppeesd feedior : . 
: z 
Fer a th catanibab ie By = 
_@+tbzta 
ual? 2246” 
and the proposed fuxion i transformed to 22; the 
fluent of which is 1. (2 = 4 6) + const. therefore 
eters at {e ce +U4+-V(a+br42")7} 
Hence, [a= 1. fete+v(2 =«)i} | 
Suppose dy = d x4/(a* 4 2*); we put 4/(a* + 2*) 
=z—2, hencedy=zdz—xdz, andy=— }xt+/fzde. 
Instead of d z, put its value 2% (a 4 2), and then, ta- 
king the fluent, and substituting, we find ee 
yactiey(ode)tiel fry yes ay} 
Ie we put dy = 77-"* under this form dy y/—1 
dx 
= ETT ae ea ne oe 
y=) 1, {ervey} rer! 
If y be an arc of a circle, x is its cosine (Rule (D), art. 
26.) and 4/(2* —1) = o/— 1, sin. y;, the equation of 
the fluents is therefore i 
shy Y—1=1. fos. y =e7—1. sin. yf. 
The constant correction c in this case must be = 0, 
boosie: wlien + 551g laa Se eee ‘Moreover we 
i =k, because t i if 4/— 1 ma’ 
either + or eyo e bein ‘the ada amber in 
ier’s i art, 12. 
Napiers System of Logarithms (art. 12.) by the theory 
yv¥—l1 : ; 
ae me 
putting the radical = z— 2, it 
> ar 
cos. y fo 4/1 sin. y= e 
cos, y—/—1.sinny=e 
yVv—t 
> 
cos. y= 2 ‘Gaw 
v—1 —yVv—1 
. ey yj one yt Pees 
sn. ¥ = 2/—1, : - 
considered these formals as one of the 
most analytical discoveries \ 
They were first given by Euler. Form diffrent matie 
of investigating see ARITHMETIC oF Stnes (29 
124. Case II. Let the radical be 4/(a phe ; 
the method cannot now Be apple’ witlnaee 
introducing imagi uantities: But in this case the 
trinomial « -{- Beans? te the profiakte? thads toro meal 
CHES (440) — $b, 4Y(h + 40) Hh b—2, 
Let them be denoted by  — «and @— 2a, and let us 
assume 
V(ap-bcmn)j=V/ {(e—«) (a—z)} =(x—«) x. 
always real, it is to be observed, that as a ..be—.2? is supposed oA’ k pihacive ‘cjuastily.= 
4 x* = b+ 4a— (b— 2x) will also be positive, but (6 — 22)! will always be positive, therefore b% -f- 4a must also 
4 
