lowers 
Methed, 
—y—" minator, and adding, their sum 
436 
dependent of z, 3B: reducing them to a common deno- 
i (A+ B)s—(bA +B 
which, com with the 
A+B=4, 6A4+¢ B=—/, from 
katl 2 
A=———— B= 
Hence the practicability of the resolution, and the man- 
ner of performing it, are shewn. 
te By redu- 
cing to a common denominator, we find 4x* + / x 4- m= 
fa hs LF orate fA? LB ae = — Bd). 
Hence, of indeterminate coefficients, 
k=A+C/=B— Ab—2aC,m= atC—Bd. 
By these nome ceed ay re Faes 
A, B, and y be determined. Exactly in the same 
esti tail. Sound, that the fraction 
ES +lz*+meztn 
@—a)"e—5) 
be decomposed, so as to be equivalent to 
= Azt+Br+C a: bo 
(2«—e)5 tmb - 
In general, Jet be an irreducible fraction, such, 
that the power of = in the mimerator is one 
less than its highest the denominator, and let 
v= Le goge past being polynomials which have no 
common factor, and in sara a ot 8 
est powers of x are p and gq. We may assume 
Aa? * + Ba? 3; 
Fo a of sae vtech ate reget 
degree of w exponent 1s —l same 
as that of U, de seseeunet sie tee fraction : 
IID and @ at themselves the roduc f two for 
a aeedite aioe ee, Oates peas an 
Ss wo prceling n thin was he propa raabe: 
may be mpd bers ac meee 
numerators 
Whatever be the degre ofthe numerator ofthe 
pierre Vy” by division, it may be brought to a 
of the fraction thus nee cases. | 
gon He the: dononsecior ead on eae 
my ight (Ke Be . x S, we may sup- 
oo A 
v= ee eS = 
rb Meh E aU i 
Q 
FLUXIONS. 
— 4" 
Examece 1. Let the fraction be — > By the 
ory gin, (Atanas) be he rt 
ee =90,. a ore tata 
— I— ves 
tea) e—r-—2>= G9 e+) 
so that we must assume 
A B 
2—42 
at—c—2 = 7-2 tT 
Hence we find A= —2= Band the rope ie 
2 2 
2—2 ) 240° 
Ex. 2. se ears fraction 
A 
be ope Gur a—x 
tion = 
1 
sat — x" 
——, by reducing to a common deno- - 
minator, and putting the numerators equal to one ano-, _ 
ther, we find 1= Alias +a(B-+ A); here we. 
must make aR hs 0, and pen arse $ which 
gives B=A= 3, and 
Yan) eee ma 
a?—xt”~ 2u(a+z) / 2a (as) 
. " eae’ B 
Ex. 3. Assuming ———~ (a=) = a4 age t ane 
we have 1 = Aa* +aa2(B+C) 4+2* (C—A—B,) 
which gives 1= Ae, Bot : Sores 20; 
La 
- oe ea) * 
‘ 
wisi geetres 
- Fd 1 1 é 
~ivGts t2e@—a) ett th 
oc mT At ee 
so that 
V=(et+prtqye+prt+”) - 
we may suppose 
ip Ac+B C2z+D 
V-Ptpetg) @+pete += 
e—xpl oA Be+C 
OMA yk he ig Tea 
Ra a 
Ex. 5. ois eee ee 
+ pat pp we ind A= C=—B= eo : 
Gaastte If V has real and equal factors, so that V= ’ 
U Awrt4 Ba P 
vo" Gm PS. 
bot tile better t0 pat 
A B TP 
Ex'6, Thies" forthe freee 08! 
thovigh it maybe decomposed into’ + cyt 
