lorerme 
. 4 
—_——— 
438 
Fill be obtained, by which A and B may be determi- 
ned,*. Let the fraction be ’ 39 TH fet 
z Ar+B 
(2—1) (#4241) ote 
Now pap ito aps ys eter 
apt Ra aay last spctina,'e bapiesas 
j= fae pa ects ry 
Fieie ee en he 
rately equal, and dividing the latter 
=+/—3, we find B Seer B; 
(4) Let us consider the fraction, when 4 = 
Arc+B * ~"A'r4 Br on 
GPP Fret) re 
Let sides of the equation be multiplied nage f 
(+ps+9)", , and observing that'V =(2*-+-p 24-9) 
let K= =5: , and we have 
KsAstB+(A's+B)(s4 p44) + &e. 
This case is the two » and 
Scar bs etlaak ix tas coe ohbOar In the first place, 
we substitute’ ey 2” whe ee etn 8 eq ion 
w+pe+q=—0, w e 4 . 
K=Aa %. The caspiailey “qed ake 
now so ta ads petite ye, gor Bh pa and 
hence Smo eqestios.ane get Sen bich Aso Reape. 
a taking the feesions of both sides of the equation, 
therefore, 
SE HAs (Ac BY(224p)} 
mandi Rs avec sony mem ode. walaet Clad 
eee in this expression, th a 
nary value of x in eqestion «4 x+q=0, t- 
we gong found, b vebedy, etal 
tw ’ - 
© equations are 7 are de- 
For example tthe faction be = Om 
assume it 
Arc+B A’ «c+ B’ 
=@oaspop + Paz49 
By reducing to a common denominator, ‘we find. 
P— 22 + 2r—3= 
A r+B-+(A'r+ B’)(2? —2 24-2). 
St y—f. Ths being cobeiated Ba'ay 2=0 is 
r=14/— being substituted for 2, e equa 
tion becomes 
4/1 = A+B —1. 
Hence the two equations are ey ee 
A=—1, therefore B= — $. Substituting the values of | 
A and B in the equation, and » we have - 
P— 2x? 4 2e=(A'r + B #— as +2); 
and taking the fluxions, and divi 
S2— 42 4 SAA Pe te ruts 
on again substituti 
29/1 =— 2/4 2(A’ BY) /— 
hence’ A’=1, reese rents iar " 
_ ALB, p, qm b 
oe Clee 
FLUXLONS. 
_ *=+3 + 
— Fare? age 
ie hs wes eure acy 
into its by the tri 
ed in Anitn! we vigns 
comprehends also ol. 
of ie cata 
Equations Nu 
vier fedétions, 068 lee I Tit, Cal. 
Post: cap. xi. eaa ¥ os 
Fluents of Rational Fractions. 
119. We have seen, that every rational fraction may 
be reduced Rens be a 
@—ap 
Arce B 
of 22 
oe, ee, two penne 
, tions are transform to 
ate Att Br 
Case I. Mets 
(Art. 114, Rule ITIL.) But But 1. (c) being a al- 
together arbitrary, ronmey pat bnedt cf Ne AGS 
+ and then the fluent. is i be 
dx id . 
P alepet amet: (art, 116,, 
dx 
And singe —— ees 
mij altas gare a rd 
g{herVe-9410} 
dx plats), oe 
e—2 a—« 9 
In like manner, @ss)de = 28 2% has for 
its ‘ent —21 Cee er 1. (@ 14 “e L(o), oF 
Goa 
SP att 
or 
2dx a 
Ads, has for its fluent 
¢ 9 regi wcearer wee 
(art, 114. Role IL) ~ 2: 
—ay~" 
. For the management of inpousible ox imeginary quantities, see ALczania, At, 190-194 
AAhis AXL (2-2) +16), | 
