47 3 
FLUXIONS. 
Cor. Hence, we may find the fluxion of xyz. For if 
zizzxyz, and zuxzxy, then v—wz, and vzzwz + zzu ; bn 2 
zo — xy, .• .wz=zxy +yx ; fubftitute thefe values for ui and 
w f and we get vxzxyz+zxy zyx. 
15. In like manner we proceed for any number of fac¬ 
tors ; hence, the fluxion of the produCt of any number 
of quantities is found by the following 
Ru le : —Multiply the fluxion of each quantity into the 
produdt of all the reft, and the fum of all the products is 
the fluxion required. 
Ex. 1. The fluxion of x*y 3 is x 2 xzyf Ey 3 X 2X *— 
3x 2 yy+2y 3 xx. 
Ex. 2. The fluxion of y 2 x s z is x 3 zyfy 2 y-\-y 2 — 
2 75 1 S', X c 2 a. 14. 
.v 3 x -f-j 2 x 3 z — - x 3 zy 2 y + —y 2 % x 3 x -\-y 2 x 3 z. 
2 3 # 
Ex. 3. The fluxion of w m x"y r z’ is mx”y'Z’zu m 1 zv-\- 
nw”yz’x H — 1 x + rzo'”x n z s y r — i y-\-su)'"x”y r z 1 — 1 z. 
Ex. 4. To find the fluxion of x 2 X By 
the laft rule, the fluxion of a^-^y* 1} is -X a_4 +-> 4 |z 
2 
X 4 j> 3 ji=r; 6 x^ 4 p^]jXj' 3 j; hence, the fluxion required 
is x 2 X 6x a 4 -\-y' , l 7 Xy 3 y-\-a AJ r.y ^lx ixx. 
Ex. 5. To find the fluxion of 
Find the fluxion of each part by the laft rule, and the 
yy 
fluxion required is \J a--\-x 2 X ^ + f b‘ +J 2 X 
17. Put z = ~, then zy: 
y 
•—vfy 
y J 
r 
Ex. 3. The fluxion of — is * ■ * A - >} * -^ 
zX xy+yx — 2xyz 
~ Z* • 
„ . r a. —ax , , . 
Ex. 4. The fluxion of-is——; for a being conftant. 
xx 2 
the fluxion of the numerator is nothing, and therefore the 
fluxion of the numerator multiplied into the denominator 
is nothing ; in this cafe therefore, the fluxion of the fraction 
is minus the fluxion of the denominator multiplied into the 
numerator, divided by the fquare of the denominator. 
Ex. The fluxion of — is 
x n 
■ nx"~ 
x 2 " x"-i~ t 
nx —”— ‘f ; or the fluxion of x —" — — nx — ” — i x-, when 
therefore the index of a quantity is negative, the fluxion 
is found by the fame rule (Art. 13.) as when the index 
is pofitive. 
Ex. 6. The fluxion of 
\/ a 2 + .v 2 
fb 2 +y 2 
a 2 + x 2 ) i x XX x 3/ b 2 + y 2 — b 2 i X yy X \J a 2 +x 2 
b 2 +y 
V a 2 + x 2 X.rj 
b 2 +y 2 If 
4/ a 2 -p x 2 
j6. It appears from this Prop, that the fluxion of xy 
confifts of two parts, xy andji-, the former part arifing 
from the increafe of y by y, and the latter from the in- 
creafe of x by x ; but if x fliould decreafe whilft^ in- 
creafes, then the fluxion, exprefling the increafe of xy 
upon the whole, will be xy—yx, being the increafe minus 
the decreafe. Hence, to exprefs the rate at which any 
quantity increafes , the fluxion of the parts which increafe 
muft be written with the flgn -p, and thofe whicli de- 
creafe with the fign —. Now the increaflng quantity is 
confidered as pofitive; but if a negative quantity increafe 
in magnitude, it muft be confidered as a decreafing quan¬ 
tity, and its fluxion will be negative. In like manner, 
a negative quantity decreafing in magnitude muft be con¬ 
fidered as an increaflng quantity, and its fluxion will be 
pofitive. If, therefore, the fluxions of increaflng quan¬ 
tities be w ritten with the fign and of decreafing with 
—, whenever the fluxion of any quantity is pofitive, it 
fliews that quantity to be in an increaflng date ; and when 
negative, to be in a decreafing date. 
Prop. VIII. — To find the fluxion of a fraction -. 
y 
r, and zy+yz — x (Art. 14.) 
— ~ *—-7-- —Hence, we find 
y y y2 ’ 
the fluxion of a fraction by the following 
. ^ uLE - ro C 1 the fluxion of the numerator multiplied 
in.o the denominator, fubtraCt the fluxion of the deno¬ 
minator multiplied into the numerator, and divide by the 
fquare of the denominator. 
Ex. 1. The fluxion of — i s UMf _ 
Ex. 2. The fluxion of I' 3 * *+> — X 3 * a s 
fXx-t-y— x + yx:f 
fa 2 +x 2 x \J b 2 +y 2 
The putting of a quantity into fluxions is called the 
dirett method of fluxions. 
18. Scholium. In queftions of a geometrical and 
philofophical nature, where we want to get the relation 
of the fluents from the fluxions, and in others where we 
want to find whether quantities are pofitive or negative 
from the relation of them to their fluxions, it is neceflary 
to pay regard to the figns of the fluxions, as explained in 
Art. 16. But in putting equations into fluxions, as in 
the Problems de Maximis et Minimis, although one vari¬ 
able quantity may increafe at the fame time that another 
decreafes, yet we may write the fluxion of each pofitive; 
for by writing it fo in each equation in order to obtain 
the fame fluxion from the different equations, the refult 
will not be altered. In thefe and fuch like cafes, we 
may therefore make the fluxion of each quantity pofitive. 
We may further obferve, that when any fluxion becomes 
negative according to the above rule, the quantity which 
exprefles its value becomes negative. For inftance, if 
r— the radius of a circle, x — the verfed fine, y~ 
the right fine of an arc, then y 2 =2 2 rx — x 2 , andj>“ 
TX _ XX 
-— -; now for the firft quadrant x and y increafe, 
and each fluxion is pofitive, and the value of y is pofitive, 
x being lefs than r; but in the fecond quadrant, y de- 
creafes and its fluxion becomes negative, and its value 
becomes negative, x being greater than r. This circum- 
ftance is fimilar to the cafe of a quantity palling through 
0 and changing its fign, for y—o at the end of the 
quadrant. 
19. When we compare the fluxions of two quantities, 
by comparing the increments that would be uniformly ge¬ 
nerated in a given time, the quantities have been Aip- 
pofed to be homogeneous, there being no relation between 
thofe which are not homogeneous; yet if, of two hetero¬ 
geneous quantities, the numerical value of one be exprelfed 
in terms of the other, it is manifeft that there will be no 
impropriety in exprefling the fluxion of one in terms of 
the fluxion of the other. If one fide of a right-angled 
parallelogram be reprefented by 6 and the other by 9, we 
fay, 6 x 9=54 the area ; our numerical operation is per¬ 
fectly correct, but no one ever imagined that the units 
reprefented by 54 are homogeneous to the units repre- 
fented by 6 and 9 ; if 6 and 9 reprefent inches in length, 
54 will reprefent fo many fquare inches, or fo many fquare 
areas, the fide of each of which is 1 inch in length. Or 
if a and x reprefent the tw<o Tides, the area of the paral¬ 
lelogram will actually be ax } referring that quantity to 
its 
