504 
FLU XIONS. 
a’'.v 
Kx”’.v I.x'"x 
-px “' 
*4- &c. a —a x—b 
K*”j . Kx" . K ax ” : — 1 
4 , &c. Now (Art. 99, 
N<2 2 —1 
quently-N c •— 2Na — o; therefore N 
4 , Sec. 4- Ka»x 
tlie fluent of —'-is 
x—a m m —1 
h. 1. x — a-, in like manner, the fluents of all the other 
quantities are found, the fum of all which is F. Now 
* __—-— x™ 
the fum of all thefe quantities=:K.-|-h+, Sec. X— 4 * 
x”’- 
ia—c 
of- 
,2 . t . 
> ■*- ■ 
X 
i; M: 
Hence, the fluent 
&c. It, 1. x— a-\-'Lb n 
Ka-J-L,6-J- &c-X- 
hl -i 
Xh.l. x—J+ See. But by Dr. Waring's Med. Alg. la ft 
edit, in the Addenda, K-f-L-*-, Sec.zzzo ; Kc-j-Li-J-, Sec.— 
o; Sec. through all thofe terms, whe n m i s lefs than n ; 
in this cafe therefore F=z:K.fi" , X h. 1 . x —a+Li”Xlt, 1 . 
If m be equal to or greater than n, the co¬ 
efficients of the firft n— 1 terms will become=o. 
103. If \m be lefs than n, the quantity — 
a—cl 
Lxx + Mx Nx 
or— -- 4--may be thus 
x 3 — px 2 +qx — r’ x—oX x—c 
found. Put x — a—z, then x—z 4 a, and xzzzz ; hence 
Lya+ M t_ Lzii+Lflg+Mz (if L«4-M=£) —+ 
x —a! 2 z 2 Z z 2 
whofe fluent (Art. 45. and 37.) is I, x h. 1 . 2 — -= L 
1 , - b Ni 
X h. 1. ^— a — •-j and the fluent of - is N x 
c 
h. 1. x — c, 
105. If two of the roots be impoflible, thofe two bino¬ 
mial fractions niufl be incorporated into one. Thus, let 
1 L M N 
■=7—7; +-—- + --, and fuppofeaand 
■px’ : 1 -J-, &c. x 3 — px 2 +qx —r x—-a ' x—b x—c 
may be refolved into 
K.v l.x M-v 
*— ax—b ‘ x—c 
4, Sec. for in tliis 
cafe K xx — bxx—cX > Sec. 4 L X*—<*Xa— c x> Sec. 
, a ” 
xzza. 
4 -, Sec, — x m ; lienee, if 
A- 
if x—b, Lz 
-by.a — c X, Sec. 
'3 &c. The reafon why 
i—a X b —c X, Sec. 
Btnuft be lefs than n in this: The quantity K X x—b 
X x—c x &c. +L X x—a x x —cX» Sec. 4, &c. —x m z=. 
o; and that this may be always true, the coefficients 
of the like powers of x muft be alfumed^zo, and by 
fuch an affumption you would deduce the lame values of 
of K, L, &c. as above. Now the product of each of the 
quantities into which K, L, Sec. are multiplied, is of n —1 
dimenfions in terms of x, there being n —1 factors ; hence, 
if m be greater than 11 —1, there is only one term in which 
x is of m dimenfions, therefore this term can never be 
made to vanifli, generally with the reft. But if m be 
equal to or lefs than n —1, then this term x m will come in 
with others having the fame power, and the whole coef¬ 
ficient may be made — o. But the denominators may be 
otherwife exprefled ; for as a -—a x a—<6x> Sec . — x n — 
px„ —t-h&c. by taking the fluxion we have a X a —b X 
x—c x. Sec. 4 x x a —a X a— c X, &c.-f, &c. z=z nx' 1 — 1 x 
— n —1. px"— 2 x + , Sec. hence, if x—a, we have a—b X 
a —c ><j & c - —na" — 1 — n — 1. par -— 2 Sec. If x—b , 
then b—a x b—c X» Sec. z=znb n — 1 — n —1. pb "— 2 4 i 
Sec. and fo on for the reft ; hence, take the fluxion of 
the given equation, omitting x, and write a, b, c, Sec. for 
x, and we get the denominators. Hence, when\m is lefs 
b to be impoflible; then.. L , M _ L+Mx2-LHMg , 
A a x b x 2 — n-\-bxx + ab 
and the impoflible quantities vanifh, as will appear by 
fubftituting m 4 it yj —1 for a, and m — n^J —1 for b. 
_ _ , > exx+dx 
Prop. L .—Let F=—- ; to find F. 
x 2 —px + q J 
106. Put x — hp—z, then a=z 4 iP-> and x—z • hence, 
dx=dz, and cxr—czz + ^pcz, .-. exx 4 dx—ezz 4 \pc 4 ~d 
X i— (if i pc+d—e) czz+ez; alfo, x 2 —px+iP 2 =e 2 } 
hence, x 2 —px+q=z 2 -\-q—\p 2 — (if q—xp 2 —a 2 ) z 2 ±a 2 , 
according as a 2 is pofitive or negative, or according as 
the two values of a are impoflible or poffible. Hence, 
ei, 
rr—7 • Now (Art. 45.) the 
F — 
czz, 4 ez 
z 2 ±a 2 
fluent of 
is ic x b. 1 . z 2 ±a*. Alfo, taking 4* 2 , 
ez e az 
— --= — X -1-: 
z +a a* z + a 
, whofe fluent (Art. 46.) is — x 
cir„ arc. rad. = a, tan. = z. But taking —cc 
■■ whofe fluent (Art. 45.) is — X h. 1 . 
z 2 —a 2 2 a 
than n, the fluent of • 
is K X li. 1 . x—a 
a'— px n —‘4, &c. 
4 L x h. 1. x—b 4, &c. which agrees with the conclu- 
fion in Art. 102. becaufe K—Ka'”, L=L^’“, Sec. 
104. If two roots a, b, be equal, one of the quantities 
rauft have a quadratic divifor x — a. For example: 
1 La4M N 
Let— ---4-: then reducing the 
x 3 —px 2 +qx—r x—a x _. c 
two quantities on tlie right to the fame denominator, 
and making the numerators equal, we get La 2 — Lcx4 
Ma — Me 4 Na 2 — sN«a 4 Na 2 — 1 — o; hence, 
making L 4 N — o, M — Lc — 2Na = 0, — Me 4 
fiJd - —- J 
Na 2 —r ==; o, we have, L = — N, M =- ; confe- 
2 a ^ 
7 CL 
--; call the fluent of this fecond part B, and F=§ c X 
z—■a 
h. 1 . z 2 ±a 2 -{- B. Call this fluent 
Prop. LI. — Let F= X — — ■ —» to find F. 
a 2 — px-\- q 
107. If the roots of a 2 — 6*4c=o be both poflible, then 
j K L 
(Art. 101.) refolve ———— into- 4 -and F 
v > -x 2 — px-\-q x—a X i —b 
— 4. whofe fluents are found by Art. 102. 
x—ci x—b 
But if the roots be impoflible, divide a"a by x 2 —p x -\-q 
until the remainder becomes cxx-\-dx, c and'^ being put 
for the coefficients which arife from the divifion, and 
let the quotient be x m — z x ax m 3 a- -4 bx m A x 4- 
Sec. where a —pi b—p 2 —y> Sec. ; hence, r —x m 2 .v 
, exx + dx 
J r e !X ”— 3 x+bx— A x+, Sec. + x2 __ p ^p J confequently 
a— t , ax m — 2 , bx’" 3 
(Art. 37. and 105 ) F= 4 * 7^ + ' 3 +» 
Sec. 4 CL^. 
If m=z 2, then F rz A4-CL., 
If m — 3, then F = ijf 2 4 «+ ( ic 
If ot = 4, then F = i a 3 4 z ax 2 -\-bx-\-(\^ 
Prop. LIL —Let F = to f nd F - 
jo8. Put a = -z=ez ~~*, then x "-—’«'+*, and x n ~~ 2 x 
z 
