FLUXIONS. 
expressions as involve two or more vari- 
able quantities, substitute, instead of such 
fluxion, its respective flowing quantity ; 
and, adding all the terms together, di- 
vide the sum by the number of terms, and 
the quotient will be the fluent. Thus, 
x v 4- a - 1 / 2 x y 
the fluent of xy -f -y x = — — - — — —~lf~ 
— x y ; and the fluent ofx y z -{- y xz-\ -xyx 
xyz+xyz + xyz __ 3xyz 
— 3 3 ‘ " ' 
But it seldom happens that these kinds of 
fluxions, which involve two variable quan- 
tities in one term* and yet admit of known 
and perfect fluents, are to be met with in 
practice. 
Having thus shewn the manner of find- 
ing such fluents as can be truly exhibited 
m algebraic terms, it remains now to say 
something with regard to those other forms 
of expressions involving one variable quan- 
tity only ; which yet are so affected by 
compound divisors and radical quantities, 
that their fluents cannot be accurately de- 
termined by any method whatsoever. The 
only method with regard to these, of which 
there are innumerable kinds,, is to find 
their fluents by approximation, which, by 
the method of infinite series, may be done 
to any degree of exactness. See Series. 
Thus, if it were proposed to find the fluent 
0 f_fL£- it becomes necessary to throw 
a—x } 
the fluxion into an infinite series, by di- 
viding ax by a — x: thus, a x~- a ■ 
+ xx , x‘ x 
j ' — r 
a 1 <r 
X X , VC X , 
u < +> 
&c. 
3 a 3 1 a 4 
Now the fluent of each term of this se- 
ries, may be found by the foregoing rules 
, 4 v . x 2 . x 3 , x 4 ,x b , 
to be * + --{- -g-j -f — -f ,&c. 
In order to shew the usefulness of fluxions, 
we shall give an example or two. 1. Sup- 
pose it were required to divide any given 
right line A B into two such parts, A C, 
C B, that their products are rectangles, may 
be the greatest possible. Let AB = «, 
and let the part A C, considered as variable 
(by the motion of C towards B) be denoted 
by x. Then B C being r= a — x, we have 
AC X BC=« — xx, whose fluxion a x 
— x being put = 0, we get ax = x x-, 
and, consequently, x — i a. Hence it ap- 
pears that A C (or .r) must be exactly one 
half of A B. 
Ex. 2. To divide a given number a into 
two parts, x, y, so that x m y n may be a 
maximum. 
Since x -j- y = a, and x m y n =. max. the 
fluxion of each = 0, the former, because it 
is constant, and the latter, because it is a 
maximum ; .'. x -{- y — 0, and in y n x m ~' x 
n x m y n ~ l y = 0; hence, x — —y, and 
I —1 fl 41 rv* /if 
; therefore 
in y" 
my 
-y- 
x:y. 
71 OC V 
z — ; or in y — n x, and m : n :: 
my ’ 
Now y 
n x 
in 
\ x 4- — a, con- 
1 m - 
sequently * = ; and y (= ) = 
n a 
m n 
If in = n, the two parts are equal. 
Cor. Hence, to divide a quantity a into 
three parts, x, y, z, so that x yz may be a 
max. the parts must be equal. For sup- 
pose x to remain constant, and y, z, to vary ; 
the product yz, and consequently xyz, 
will be greatest when y = z. Or if y re- 
main constant, the product x z, and conse- 
quently y x z, will be greatest when x —z. 
Thus it appears that the parts bahst be 
equal. And in like manner it may be 
shewn, that whatever be the number of 
parts, they will be equal. 
Ex. 3. Given x -|- y -}- z = a, and x y 2 z 5 
a maximum, to find x, y, z. 
As x, y, z, must have some certain deter- 
minate values to answer these conditions, 
let us suppose such a value of y to remain 
constant, whilst x and z vary till they an- 
swer the conditions, and then x -f- x = 0 
and z 3 x -\-3x 
3x.Z A X 
^ z = 0 ; hence, x =. - 
3xx 
, ,\z — 3 x. Now r 
z 3 z ’ 
let us suppose the value of 2 to remain con- 
stant, and x and y to vary, so as to satisfy 
the conditions ; then x -\~j = 0, if x -f- 
2 x y y = 0 ; hence, x 
. _ 2 x y y 
y — 7^ 
— ■ — ifLl ... y — 2 x ; substitute in the 
y 
given equation, these values of y and 2 in 
terms of .r, and x-\-2x-\- 3x=za,or6x=z a 
1 1 X T 
hence, x = - a ; .*. y = g a ; 2 = - a. In 
like manner, whatever be the number of 
unknown quantities, make any one of them 
variable with each of the rest, and the va- 
lues of each in terms of that one quantity 
will be obtained ; and by substituting the 
values of each in terms of that one, in the 
given equation, you will get the value of 
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