FLUX 
e 2 XX+2X 3 X a-X 7.X-X • • . 
— - —— ■ - - - — — a 2 a-i-2F ; hence, — 
4 /a 2 x 2 +x 4 fa 2 -' r x 2 -fa 2 -\-x 2 
T—^ic — %a 2 v, and F —%w — %a z v. Call this P. 
. .9 • 
I O N S. 
OX 
V2ax —x 2 
Prof. XL.— Let F = 
x 2 x 
505 
, ——— — ./-F ; hence 
y 2ax —x 2 V lax —.v 2 
•w, and (Art. 46.) F —z — w, z being a 
92. AflTume v — 
4 /a 2 —X 
ax 
to find F. 
, then (Art. 46.) v — 
4/ a 2 —x 
cir. arc, rad.=a, fin. = x; put w =4 /a 2 x 2 —x 4 , then 
• a 2 xx —2x 3 x a 2 x 
X 2ax — x 2 
cir. arc, rad. z=.a, verfed fine — x. 
Prop. XLVII. Let F=-Lf., to find F. 
x—a 
99. Divide the num. by the den. till the index of * in 
the remainder =0, and the remainder will then be a m x ; 
4/ a 2 x 2 - 
4 /a 2 
2\ 2 x . . Fence, F = x m i x-\-ax n 2 x-\-a 2 x n 3 x-\~, &c. a 
- =zav — 2 F ; x 
4/a 2 — x 2 
hence, F —\av — ±w, and F =±av — \w. Call this Q^_ 
Prop. XLI .—Let F— to find F. 
4 /a 2 -{-x 2 
ax 
93. Aftume v=z 4 / a 2 x b -\-x 8 > then v 
" " ' ■ v.8 
X - : therefore (Art. 37. and 45. F =2- 
x—a m 
_ - -j-, &c. +a m Xh. 1 .x— a. Here m mtill 
m —1 m—2 
be a whole politive number, othervvife the index of can¬ 
not become =0. If the denominator be xfi-a, the terms 
2a 2 x 5 x-\-4.x , x will be alternately -J- and —. 
- z rm - • 
3 a 2 x 2 x 4 x 4 x 
4 /a 2 -fix 2 4 / a 2 -\-x 
hence, F=au~— P, and F——Al_ p, 
4 4 
2 . 
4/ a 2 x 6 -J-x 
(Art. 88.) 3a 2 P + 4F j 
3a 2 
100. Prop. XLVIII— Let F 
bz 
+a^z ,r_1 ^ ^'Xz r ' 
a -j- bz" 
to find F. 
Prop. XLII. Let F= 
X 4 X 
a 2 —x 2 
, to find F. 
« fy% . ■ - „ , • 3a 2 x 5 .v—4 x 7 x’ 
94. ACTume v—\/a 2 x < * —x 8 , then v .~— — 
V a 2 x 6 —x 8 
(Art. S 9 .) 3a 2 Q—- 4 F j 
3a 2 x 2 .*: 4x 4 x 
4/ a 2 —x 2 V a 2 —x 2 
hence, F—-- and F=— Q^a. 
4 4 
In this manner you may continue the fluents when the 
n umerators are x 6 x, x B x, x*°x, 8 cc. by affiiming v~ 
4 /a 2 x 10 ±x* 2 , 4 /fx^±x^, 4 /a 2 x' 8 ±x 2 °, See. refpec- 
tively, anil by taking the fluxion, you will, in like man¬ 
ner, get v in terms of the given fluxion and of the next 
inferior fluxion. 
Prop. XLIII.— Let ¥—x"x\/a 2 ±.x 2 , n being an even 
number , to find F. 
95. Multiply and divide the fluxion by 4/a 2 ±x 2 , 
* CL * X^X *h 2 £ 
and F = — — ; hence, as the indices of x 
V 
in the numerator are even numbers, the fluents of 
a 2 x"x x’‘+ 2 x 
—r> an d - may each be found by the me- 
b 
—- b w 
a 
—7X2" 
b 
—m 2 "‘ 2 Z + 
&c. 
b 
X 2 " 
. y 2 n ley- 
b 2 ~ 8 cc. 8 cc, 
continue this divifion till the index of z in the remainder 
Qfi i 
becomes m— 1, and the remainder will be be 
-p, See. ck ■ 
X 
> =- X 
a 
z m 1 
b r 1 afi-bz™ 
mbz'"'~' l % 
X 
b'~ 
— X 
- 1 
•- 2 
■4. 
a r 1 
X 
mb' 
rnr~ — rr. 
afi-bz” ” 
a z rm — 2’" a r 1 
-X -- +> — —7— X h. 1 . a-}-£.z’\ Here 
b 2 rm —2 m inb r 
r mull be a whole positive number, otherwife the index 
of z can never become m —1. 
1 K L 
Lemma. — Let ———7—=-+- - + 
/ a 2 - 
4/ a 2 ±x 2 ’ 
4 / a 
tliod direifted in the laft article. If n be an odd number, 
F may be found by Art. 41. 
Prop. XLIF. Let F —xfzax —x 2 , to find F. 
iff 96. Let the radius AO—a, AP=x, 
then the fine PM=i / zax—x 2 , tlierefore 
F =xf 2ax—x 2 — (Art 49.) the fluxion 
of the area AMP; hence, F = area 
APM. 
Prop. XLV. Let F —xxi/2ax — x 2 , to find F. 
97. AlTume za — \ X 2 ax —x 2 | 2, then zu~ax—xx x 
V 2 ax —axf xax —x 2 — F ; hence, F=ax’ f 2 ax — x 2 
—zo, and F—ax area APM— w. 
• v y» 
Prop. XLVI. Let F— to find F. 
y 2ax —x 2 
M 
x" — px n *+, & c. x— a x— b 
&c. to find K, L, M, &c. where, a, b, c, 8 c c. are 
x— c ' 
the roots of x" — px n 1 +, &c.2=o. 
101. Reduce the fradftons to a common denominator, 
and it will be the fame as the denominator on the left, and 
confequently the fum of the numerators =1 ; hence, 
Kx.v-iX*-cX. &c. + Lxx-flX^-cX, &c. + MX 
x—a X x — b X t 8 cc.-\- 8 cc.—i ; now as this is true let x be 
what it will, make x=«, and then Kxc— by.a—cy., &c„ 
f 
Make x—b, and then L 
1 
’ a — by.a —i’X,&C, 
Xb — axb — cx, &c. = 1, .*. L: 
In 
98. AflTume w — 4/2ax—x 2 , then zo — 
ax — xx 
4 / iax —x 2 
b—aXb — cX> See. 
like manner, we get other numerators. 
Prop. XLI X.—Let F=^,_^._ 1+ &( T, to find?, m 
being a whole pofitive number. 
t02 - Let x'“x&c. = ^+^I+ &C - thea 
K, L, &c. are known by the laft article 3 hence, 
