r; — z 'a; hence, F — 
i x'W 
q X i /x 
-[-x 2 
FLU X I O N S. 0 O 5 
_-v"' 2£ __ _ x"x the e quation (A), and omitting x, we have (B) n H x 
l—px + qx 2 Jtt«:“‘+,iKx 44 1 x~x+tf + K &c. 
x ~ x —°> makexir;— and wehavesHxc—a" ’-]-K x 
= ( lf L = q', ~—P') — ~ X b ~ a — o ; hence, K = — T~ — — == 7 T+»; thus by 
( 1 1 , b a b —<2 
continuing to take the fluxion of the laft equation, and 
x 2 -p'x+q 
which is the fame as the laft form. 
then making * — — a, we fliall get tiie values of L, &C. 
In like manner, if we make xfi-bzzo, or x =—we find 
P=r=r^; then by taking the fluxion of the laft equa- 
a-—b'‘ 
Prop. LIII.— Let i= -== , to find F. 
zf a -\-cz ’ 1 
. i ^ f a\. .. ... — m 
109. Firft F — — ■ ~ ( putting d- == - }> tion, and making x ~ — b, we get Q —~-,„+i : and by 
3/ c z\/d 2 -\-z n \ cJ a—b 
proceeding as before, we get R, &c. 
= \ n Prop. LVI.-ZeuP —--I 
x-i-a X-xfi-b* 
whole poftive number. 
X Yj z ^ l ' 1 1 
put zfizzx, and then z"=x 2 ; alfo, - — 2 - - - 
-, to find F, r being a 
z 2 x z . 2 
X X - = - ; hence, F — - v , 
z. n x z - X 
x 
n-fi c x\Zd 2 -\-x 2 
X - z~= } and (Art. 45.) F 
D , a , 4 Hx r .x Karr* . 
113. By the laftar cle T F —-- ^ 4., &c, 
x-(-a" z 
X(/^ 2 -j-X 2 
VJfifi-xf—d 
y/d 2 -fix 2 fi-d 
If d 2 be negative, F —- = x 
X h. 1 . iVx Qx'a- 
ndy/c 4 ~ ——7; 4 * =—,._rr 4 ~> &c. Put x4-«=z, then x—z- 
2 x x-\-b x-\-b 
d 2 x 
~ —42 therefore x'+i—.s-./-*- 1 and x r x—z—a X hence, 
n\Jc x ■y x —a . _ 
7 X' 
nd 2 3/ c xi/x 2 — d 2 
arc, rad .=2 fecant— x. 
, and (Art. 46.) F 
X cir. * 4 -a" 
z — a X g ~ z r — m z — re:z r — m — 1 a 4 * r. 
Prop. L 1 V. Let F:n—— — 
z ~ V a 2 -\-z' 
= to find F 
~ . u z . , ,-, - --- 
no. 1 ut x— , then x — --hence,- - x ner, the fluents of the other terms are found. 
* z 2 a 2 1 
Prop. LVII.— Given A the fluent of efi-fxf X xpx, to find 
nd 2 W c 1 2 • 
a-z r —— 2 z—, &c. where the number of the terms = 
r+i, and the fluent of every term is found by Art. 37. 
except that term where the index of z is—1, whole 
fluent is found by Art. 45. and the fum of all thefe mul- 
t tiplied by H, is the fluent of the firft term. In like nian- 
— z i ; therefore F 
1 
-2 X 
l ffi a 3 X B the fluent oje-\-fx'^ X + "x, and C the fluent of m '!' 1 
V + ^7 X’ xpx. 
114. Airume Q= e+Jx J+ 1 _]_*/> + 1 , (hen —p 4 -i 
■■ . . r ; hence, (Art. 39.) F=— ~ 2 X V x 2 +a 2 ' V.e-fijif j ra + 1 X xPx+m-f 1X «/X cf-fx n )™ X xp+*x = p+\ 
\/x~X.a~ X C + w + 1 Xw/X B ; hence, by ta king th e fluents, 
ziy -z 2 ^ /+ ■ X C +vi -f 1 x n/xB ; “Alfo, e + fix ’’]" !+1 x xpx — 
Prop. LV. —Let F —- • y . ,.— —-p —. . \ . . . _ - : —. 
efJx' X m X xpx—e X e+fx"l m X xPxff X e-b/x’-j" 
ni. Put *=tA 2 — f 8 then z-7zzc 2 —x 2 , therefore zi, >^xP J t"‘x, that is, C—rA-f^B, t heief ore C— eA-f-/B. Now 
= -**. and V ~~ ^ -V b2 +^ = ( if * 2 = / 2 ± f) from the firft fluent, B= Q 7 ~^ + - - XC , and from the 
a 
0 ] 
? A 
\/a 2 — x 2 ; hence, F = — x 3/ ,< 2 — x 2 . 
Now let AN be a circular arc whofe center ^ 
is O, and PM be perpendicular to AO, an d fecond, B = -— ; hence, 
put a—Oh, x—OP, then PM=p / a 2 — x 2 • f m-\-i X nf 
hence, F = — the fluxion of the area QJ"M 4 -i xneh. , 
OPMN (Art. 49 .), confequently F=— ^-j -- "> con f e< l u ently B 
m -\-1 X nf 
Qc-T+iXC _ C—cA 
C—eA C 
Lemma, To refolve 
into , 
H 
7^3 x^'‘ x fi-d 1 
K L P 
- + — 
p+i +m+i Xn 
eh CLJ-w+i X neh g ^ 
f f 
. Hence, vre may continue 
x 4 -<z 
R 
x + a I — : 
+ , &c. + 
—^ “TTT 7 by n 
■x-\-t/l x-hi b ’”— 1 1 
f p-\-i +w+i XnXf f 
the fluent as far as we pleafe, increafing tn by 1, and p 
+6 > 
_2> & c - continued to m ana n quantities refpedt 
ively. 
112. Reduce the fractions to a common denominator, \fa~+x 
and make the numerators on each fide equal, and (A) H _ 1 „ r x 2 x 
— TT" . —T7" v —- t ~,n ——2 B the fluent of- - 
X x + b + K x x-\-b x x + a + L x x + b X x+a + ifi a 2 y.x 2 
Let e=a 2 , f— 1, m=. — p— o, n— 2; then A 
— > ar) d h — h. 1. x-t”y/x 2 ^ Art. 4 5*) 5 henccp 
=£xx« 2 4-* 2 J 2 —|^ 2 A, as in 
&c. + Px A+a + Q^X xfi-a x.r+i-fRx*'+« X x-\~b 2 Art. 91. alfo, C the fluent of « 2 4 -x 2 j ixx=z ixx« 2 +x 2 '^ 
+, &c.-i. Make *+«=o, or* = —a, and every term +U 2 h. 2 
where x-^-a enters, becomes=o; hence, H x xf-b n ~i. 
_ t Prop, LVIIL— Let F zz.vx n x t where vzzh. L —-—, .tc 
or H x b—a— 1, H —-Take the fluxion of p 
b—a* J ina r “ 
1— x 
