50G 
P L U X 1 Q N S. 
V *V • 
ii?. Affume —-+ r=F, then vx'x 4 - --[- r — 
n-\-1 ?2+t 
—= (by divifion) — 
n -\-1 x i —x n-f-i 
X 
where the law of continuation is manifeft, and the ferie.a 
will terminate when m is a whole pofitive number. 
Prop. LXI.— LctY— «W, to find?. 
• • X n "^‘ ^ V f • X \ » _ 
F = vx'x ; hence, r =- = ( becaufe v = - ) I I8 * Aflame F — a x x px-\-qx n —‘-J-rx:"— 2 +, See.' 
n + 1 \ l—xy and let m=zh. 1 . a ; then (Art. S3.) maxx is the fluxion 
»+t+ ■ 1 of a-r ; hence, by taking the fl uxions, 
m^xxpx ^-qx — 1 + rx’~ 2 +, &c . 1 
. « r X npx »— < x+n— ,. qx"—* i+, &c. J ~ & * * * 
divide both Tides by a c x, and tranfpofe x\ and we have 
mpx n -\-mqx n — x +_ mrx n ~ 2+, gee .'| 
—•*’ ! + 72/X>‘—1 + 72_i. ?J r —2 +, & c . j - ~ ° ’ 
hence, mp— 1—o, + np — o, wr-|-«—1. q—o t Sec. .•» 
n P 72 n —1 .q 
™ _J - TO 2 ’ r ~— = — 
72.72-I 
3 > & c * therefore F = ar x 
-x"x — x" —*x—, See. 4 --—, therefore r — —— X 
’ ~ t—x «+» 
A'”+* X" vx”^ 1 I 
w+1" + ~ n X > & c *— v > bence, F = ^7* + W+I X 
a '' +1 x 
--(-> &C. - 
W-j-I 72 
Prop. LIX.— Let F=»x”-Kv, where v is a circular arc whofe 
radius is 1 and tangent x, to find F. 
, . _ vx ”+ 1 1" 
11 6. Affume —■— + F : then vx’x 4 - __ _ , 
"-H 72+1 + 
r — F = vx n x. Let 22 be an odd number, and then r=. — 
*.+ ‘® . , . . *" +1 * 
rrt-— (Art.46.)— — —- = 
"T 1 72+1 X 1 +X 2 
7= 
72 - I X- 72 
I 72 
— X"-- x”~ 
m m 2 
x ’ 1 i x — x"‘ — 3 d-+, &c.dtu 
0 +* 2 )’ 
72+1 
where the fign 
72.22 - 1 
'~~~s —*" 2 — Sec. where the law of 
continuation is manifeft, and the feries will terminate 
when 72 is a whole number. 
- z’z 
Prop. LXII.— To find thejluent of ... 2 " . given the 
idzz" 1 0 
* . . 72 — 1—1 
of v will be-}-or—, according as—■—- is even or odd5 
2 
1 x n x u — ^ 
hence, r— - - x—— +~-, &c. q: »; there- 
72 + 1 72 72 —2 
vx ”-\- 1 
fluent of — 
119, Affume 
az r + 1 
fore F =- 
+ 
X 
X" . X"— i 
-, &c. +: v. If 
72+1 ' 72 + 1 ' ' n ■ 72 -2 
h be an even number, the laft term of the divifion 
1 _ h ~ + QJ"or the fluent; then, by tak 
• n ■ , ?-+1 Xtfz'zX i±:*“ naz r +’ , z 
ing the fluxion, w’e have ■ 
z r i z r z 
t±z "| 2 
+ 
I +x 2 
- ZJX”+ 1 I 
= ± h. 1 . + 1 +x 2 ; hence, F = —:— + 
72 + 1 72+1 ^ 
°-= t ±rv 
+ Q^ but 
I - . 7222Z r +’ , 2; 
X r +iX«i r «+ 
I dlZr, 
idzz‘l J 
7 iaz r ^«i, . . naz’i 
: ———■=—-—— ; hence, 
i±z" i zdz” 
1 dzz* 
z'z 
x n x n * - 
--1-, &c. ±; h. 1. 1/ i+x 2 , where the fign 
72 72 2 
of the lad term is + or —, according as |t 2 is odd or 
even. 
Prof. LX .—Let F —z m x r - — x x, where z—h. 1 . x , to 
find F. 
117. Affume ¥—az"'-\-bz m — 1 +ez m — 2 + , &c. a, b, c, 
&c. being variable coefficients in terms of x ; hence, 
by taking the fluxion we have. 
1 —— . . naz r z 
X r-\-iXaz r z—naz r z + —- + o — 
‘ — 
1 ±z n 
z r z 
naz r z 
r+iXfi —na X ~~ + + I _ l _+ |2+ affume na=x, or 
z’z 
naz’z 
a — -, fo that the terms and ma y deffroy 
az m +ia'” — 1 + cz m 2 +, 
mazz m 1 +772 —1 ,bi,z m — 2 + , 
but by Art. 45. z — ~ ; hence, by tranfpofition, 
> 1 =:s;" , a;’ ! — x x ; 
-,&c.J 
r+1 z’z 
each other, and we have Q= 1-x-; hence, 
^ a \±Z n 
Z r Z r 4 -x 
if P be the fluent of —3—, we have Q = 1 —-x P; 
-L- yl% r> * • 
I ~zzz 
72 
az m *\- bz m — 1 + cz m — 2 +, See. j 
— X” 1 xz m -j- z m —1- 
772 - 1 .bx-K - 2 
I Z’\ X 7*4-1 
confequently the fluent required is - x + x __ —— 
+, See. 
therefore a — x" — 1 x=o, i+ — o, c + m ' -^ x — 
x" 
X 
max 
o, See. hence, a—x” — x x. .’.a— -—; b : 
n x . 
-mx *— 1 x 
b — 
— nix ’ 1 
72 ~ 5 
-772 - 1 . id- 
72 * 72 
■ 772-IX-- i X' 772.772- I. X” 
--r-r, C — ---; Sec. hence, 
72 ■= 
22 J 
_ X* PIX" . , , 722.772 - 1 . X 
F x — X X 2"—-X 2 — 2 —, See. 
X P- 
Prop. LXIII .—To find fluents where there are two va~ 
riable quantities in the given fluxion. 
120. It frequently happens, that a fluxional equation 
contains two variable quantities, in which cafe, they muff: 
either be feparated, or reduced to the fluxion of fome 
known fluent; but no general rules can be given for this 
purpofe, and the reductions muff be left to trial and the 
fkill of the analyft ; the following Rules, however, may 
be of fome ufe. 
Rule I. Multiply or divide the given equation by 
fome function of the unknown quantities, fo as to bring 
them to a form whofe fluents may be found by fome of 
the 
