507 
FLUXIONS. 
the rules already given, or to the fluxion of a known 
fluent. 
„ „ x . v ax m x 
Ex. i. Let— - 
* y y‘ __ _ 
nx”y n , and it becomes ny"x n i x-\-nx"y n y ~ nax m r n x j 
now the fluent of the fir It part is known from Prop. 7. to 
be x"y", and the fluent of the other part is found (Art. 
-57.) to be 222-_• hence, the equation of the fluents 
nax m + ’ ,J r * 
is x n y n ~~ —r—:— * 
Ex. 2. Let x- — xz 2 ~fz 2 . As z does not enter info 
this equation, in deducing the fluxional equation from the 
fluent, £ mud have been fuppofed conftant. Multiply 
by x, and xx — xxz 2 z=fxi, 2 , and as 2; is conftant, the 
x 
fluent is &x 2 — £x 2 z; 2 =fxz 2 : hence, z—-r====. 
V 2 /X+X 2 , 
-J-, See. when therefore x is Cendant, the value of *2 is the 
Multiply both fides by f ame as— Ch. whenv is conftant; hence, if in any fluxional 
x 2 
y 
equation in which Sc is conftant, we fubftitute for -? the 
quantity —or for y the quantity -4—, we (hall trans- 
X x 
form the equation into one in which y is conftant. By 
this fubftitution, the fluents of fluxional equations may 
fometimes be found. 
Ex. Let xy — xy—ay o, which being fuppofed to 
have arifen from fome fluent, x is conftant, as x does not 
enter. Subftitute —■ yX - lory (in which cafe y becomes 
x 
whofe fluent (Art. 45.) is z = h. + \J 2 fx-\-x 2 . 
Rule II. Sometimes the fluent may be found, by the 
addition of a new variable quantity. 
Ex. Let az—zx — xx. AfTume z=a-j-x-\-v, then z=,x 
4-y; hence, by fubftitution, ax-\-av—ax-\-xx-\-vx — xx, 
• CIV 
therefore, avczzvx, or x =—; hence, (Art. 45.) x=zay. 
V 
h. 1. v ; confequently z=a- 3 r v-\-ay.h. 1. v, and by fub- 
ftifuting for v its value z — a — x, we get x = a X h. 1. 
z — -a — x. 
Rule III. The fluent may fometimes be found, by 
firft putting the equation into fluxions, making one of the 
fluxions conftant. 
dx-^p^yx x v 
Ex. Let -^— ~x+y —f-. Make y conftant, and 
y x 
d-X^y SC x 
put the equations into fluxions, and — 4. x ~ 
__ y 
' . xyx—x 2 y a+yX’x xyx —— 
*+J+ v - 2 - - 5 hence, —r- = —, and a+y X 
x 2 =.xj> 2 , confequently x 2 x — a-\-y\ %y ; hence, 
(Art. 37. and 39.) we have 2 x 1=2 
Rule IV. If only one of the variable quantities (x or 
y) enter, fubftitute for one fluxion, the other multiplied 
into a new variable quantity. 
Ex. Let yy 3 x—ax 4 2ax 2 y 2 -\-ay*, where x is wanting. 
Afl'ume zyzzx, and we get yzy 4 z=,az 4 y 4 -\-2az 2 y 4 Aray 4 , 
d 
oryz~az 4 + zaz 2 ffia ; hence, y—az 3 -\-2az-\-~, therefor e 
y~isiz 2 z-\-%az—-— i , confequently x—zy~%az 3 £-\-2azz—~ 
—, whofe fluent is xzz^az 4 f-az 2 — -rzXh. L 2; and if in 
z 
this equation we fubftitute the value of z in terms of y, 
found from the equation y—az 3 -fzazffi-, we fhall get x 
in terms of y. 
Prop. LXIV. —In any jluxional equation if the fecond or. 
der, where the fluxion of one of the variable quantities {x) is 
xonjlant ; if fory we fubjlitute , the equation will be tranf- 
farmed into one in whichy is conjlant. 
1211 For fuppofe the value of y to be exprefled by a 
4 £*+« 2 +dx 3 +, Sec. then ^ = b+2cx+3dx 2 +, See. 
V 
Make x conftant, and take the fluxion, and - = 2c.v-j- 
conftant), and we get xy- 
;0, or X i 
xxy 9c z y , . ic' 
f-xxf-ax -— =0, w'hofe fluent is xxffiax——, which, 
as the fluxion is =0, muft be equal to fome conftant 
zbxx 2 abx 
quantity ; let it be cy, and then 
whofe fluents (Art. 45. and 46.) are y~by.L-\-ay. f 
(T ~b 
X A, where A is a circular arc, whofe radius is x and tan* 
mmmm yX 
'Qdxx^-f'Scc. Now make j. conftant, and + 
gent “7=, and L=h. 1 . ibc^-x 2 . 
V 2 be 
Prop. LXV .—To invejligate a problem by the method of 
fuxions. 
Ru les. —.1. Let all the quantities be denoted by proper 
fymbols, as is explained in the notation of fluxions, and. 
let fome one of the variable quantities (with which the 
others may always be compared) be fuppofed to increafe 
uniformly : and this may be called the principal variable 
quantity. Then the given equations, or fuch equations 
as are deduced from the conditions of the problem, muft 
be turned into fluxions, fecond fluxions, &c. in order to 
get as many equations among thefe fluxions, as you have 
occafion for. 
2. But becaufe fometimes fome doubt may arife about 
the figns of the fluxions; cbferve that any fluxionary 
equations, deduced from the equations of curves, or 
from any given equations in the problem, will contain 
the fluxions with their proper figns. But in any pro¬ 
portions made between fluxions or moments, as in fimilar 
fluxionary triangles and the like; then the fluxions or 
moments of quantities that decreafe muft be actually 
made negative ; and thofe that increafe muft be written 
affirmative : or, which is the fame thing, (fince one part 
of any whole increafes whilft the other part of it de- 
creafes, therefore) inftead of the negative fluxion, you 
may take the proper fluxion of the increafing part of 
that quantity. 
3. Since velocity is always meafured by the fpaces 
uniformly defcribed thereby; fo may the fluxions be 
meafured by the moments uniformly generated by the 
fluxions. Therefore the moments (uniformly generated, 
or, which amounts to the fame thing, confidered as arifing 
or vanifhing) may be put for the fluxions, and the refit it 
will always be the very fame in all operations. And 
fince in many cafes, efpecially in geometrical problems, 
the reafoning and calculating with the moments will be 
more eafy and evident than with the fluxions ; the equa¬ 
tions that are gained thereby muft at laft be changed into 
fluxionary equations, by fubftituting the fluxions inllead 
of the moments, which muft always be fuppofed to be 
taken in the firft inftant of their generation ; or, at leaft 
when the operation is over, thefe moments muft be fup- 
.poi'eti 
