FLUXIONS. 
477 
never arriving at the latter, as the quantities vanifh at Rule: —Multiply by the index, diminilh the index by 
the inftant that fuch a circumftance is about to take unity, and multiply by the fluxion of the root. 
Ex. i. The fluxion of x 3 is <)x 3 x_. 
Ex. 2. The fluxion of jy 5 is i sy^y. 
Ex. 3. The fluxion of -yf is —y f ji 
2 14/ 
Ex, 4. The fluxion of—XT*: 
9 99 
Ex. 5. The fluxion of 4 nj 1 ; s U.Vfv. 
7 x °3 
6 y, 
tW: 
- Ji£_. 
" 99 *TT 
place. 
Prop. III. —If theJluxion of x he denoted by x, thefuxion 
of ax will be ax. 
9. For if x increafe uniformly, ax will alfo increafe 
uniformly, and a times as ,faft; hence, by Prop. 1. the 
fluxion of the latter will be a times greater than that of 
the former, or it will be ax. 
Cor. Hence, in taking the fluxion of a variable quan¬ 
tity multiplied into a conflant one, the conffant multi¬ 
plier is retained. 
Prop. IV.— The fuxion of xdz a is x. 
,0 For « being conflant, and only connected to * by roo 7 is a , +x ., and its fluxion hence, the fluxion 
the (igns + or —, it cioes not affect the increajc or decreaje ... - — . — ; - , . 
of the quantity ; therefore the fluxion is the fame as the ie( l lllled ls 3X« --r*VX 2 xx—a^-fx^l-X 5 -v.v. ^_ 
fluxion of x, or it is x. Ex. 7. What is the fluxion of or of a--\-x 2 \i ? 
Cor. Hence, conflant quantities connected to varia- Here the root is c 2 -f-.v-, and its fluxion i.v v; hence, the 
ble ones by the figns -j- or —, difappear when the fluxions. 
are taken. 
Prop. V. —Given (x J the fluxion of x , to find the fuxion of 
x”, n being a whole number. 
11. Let x increafe uniformly by v and become x+v, 
then will x” become x -j- v b ; but by the Elements of Al- 
Ex. 6. What is the fluxion of & 2 "-j-.v 2 ) 3 ? Here the 
fluxion is -Xa'X-v"] 
2 
X 2 ,vi - 
c 2 4-v-ii 
gebra, p. 294. x + nt =zx" + nx" i v+n.- 
-x”~ 
*v + , 
Ex. 8. What is the fluxion of Here the 
root is x 2 -{-y 2 , and its fluxion zxx-\-ivy ; hence, the 
fluxion required is -xr x 2,v.vfl-21 j— 3Xa 2 -hr^2 
X x-j i-\-yy. 
Ex. 9. What is the fluxion of ,r -j-jp) ! ? Here the root is 
x-\~ y, an d its 11 uxion ; hence the fluxion required is 
2 X*+7X-v+j. 
See. and if from this quantity we take x ", there remains 
_ n —1 _ 
nx n i v-\-n. —-— x" 2 <y 2 + , See. for the cotemporary in¬ 
crement of x” ; but although x increafe uniformly by v, _____ 
x" does not increafe uniformly ; for if in the increment of Ex. 10. What is the fluxion of a 5 -J-\ 5 12 ? Here the 
x” weT'ubftitute 1, 2, 3, See. fora, and take the differences root is a 5 -\-x s , and its fluxion sv 4 .v; hence, the fluxion 
of the refults, thefe differences will not be equal; hence, 4 . 
to get the ratio of the fluxion of x to the fluxion-of x’• we required is - X<2 5 4 -a 5 [— 2 x 5x 4 L: 
muff, according to Prop. 2. take the limiting ratio of the 2 
increments. Now the increment of x : the increment of 
x'' :: v : nx" i v + n. - x n ~ 
2 
2 v 2 + , See. :: 
_1 + n. 
Ex. 11. What is the fluxion of 
2 Xa 6 -|-.v 5 !&• 
1 
Hi? This quan- 
— fx n 2 v + , See. and to get the limiting ratio of thefe 
increments, we muff make v— o, in which cafe the ratio 
becomes 1 : nx “ *, which therefore expreffes the ratio 
of the fluxion of x to the fluxion of x "; but x denotes 
a*-\-x- 
tity becomes and the root is whofe 
fluxion is axx ; hence, the fluxion required is —-x 
a- -f-x“ 
-— 1-4 
X 2XX■— — 
In like manner, bring 
9 X a .+*. 
the fluxion of x, therefore nx" 'i reprefents the cotem- any quantity from the denominator up to the numerator. 
porary fluxion of x". 
If n—o, x"—i a conffant quantity ; therefore by Art. 3. 
Cor. 3. it has no fluxion. 
Prop. VI .—To find the fluxion of x”> m and n being any 
whole numbers. 
12. Put y—x", then y m —x n ; hence, by taking the 
71 X n * X 
fluxions my" ! f—nx" 1 x, . ’ ■ j =2 (by fubfti- 
my m ~ 
r ■ , . „ nx" - * 
tilting for y its value in terms of x) - = - 
by changing the flgn of the index, and then proceed by 
the rule. 
Ex. 12. What is the fluxion of ax- +by 3 -\-cz 4 f ? Here 
the root is ax 2 A r by 3 fl-cz* , and its fluxion zaxx-fflyfi-fl 
4cz 3 z; hence, the fluxion required is -y.ax"-fiby 3 f-cz^ 
__ 3 
X 2axx-\-T > byf-\-\cz 3 i.. 
13. What is the fluxion of f • E ut K ~ 
f x*-\-\J then ss 2 — x"-> rx / f-y-y now the fluxion 
of s/a 2 -{-y"-, or of a 2 fy | s - x 1 2 X V)'— 
= - X at 
m 
Cor. Let the root be a compound quantity as a m -]-x m , 
to find the fluxion of Put y—a m + x : ’|", then y 
mx' 
ny 
a 2 +y 2 <~ixyy; hence, 22; 4— sX.vji’, there- 
£ ^ _ 2 xx-\-a"--\-y 2 ] iX..VV_2.vi--^- ( r-’ -p.vN h X9'V» 
2Z a\J x'-\\J a.-fi-y 2 ■ . 
Prop. VII.— To find the fuxion of a produB :y. 
_ „ M 14. The fluxion of xfy 1 , by the laft r ule, is 2X A ‘ J r>’ 
and n r 1 j= mx ’ n hence, j— ————— — x txx+zxy+yx+zyy ; alfo, x+y*=.x*+2xy+y*, 
whole fluxion is 2xxfi- the fluxion of 2xy-\-2yy ; make theie 
two values of the fluxion of v-|-y 2 equal to each other; 
omit the nrft and laft terms which are common to both, 
_ _ and we have the fluxion of2xy—2xy-\-2yx ; hence the fluxion 
«”' + x'"),i 1 ymx m 1 x. 0 f X y \s xyfi-yx. Otherwife thus : If we fuppofe x con- 
13. Hence it appears, that whether the root be a fun- ftant, the fluxion of xy is xy by Prop. 3 ; and ii we fuppole 
pie or a compound quantity, the fluxion of any power y conffant, the fluxion \syx ; hence, it neither be conflant, 
thereof is found by the following the fluxion is xji+yx. 
Vol. VII. No. 442. 6 F Cor. 
mx’" 
l x 
, 1- 
= - X a"’+X’ 
ny.a 4-x m | » n 
X ntx ,n 1 x — - x 
n 
