FLUXIONS. 
of this part of fluxions is comprized in these 
rules. 
1 . To find the fluxion of any simple va- 
riable quantity, the rule is to place a dot 
over it: thus, the fluxion of x is x, and 
of y, y. Again, the fluxion of the com- 
pound quantity a' -f- y, is x -\-y ; also the 
fluxion of x — y, is x — y. 
2. To find the fluxion of any given power 
of a variable quantity, multiply the fluxion 
of the root by the exponent of the power, 
and the product by that power of the same 
root, whose exponent is less by unity than 
the given exponent. This rule is expressed 
more briefly, in algebraical characters, by 
n — 1 _ n 
mx x = the fluxion of x . Thus the 
fluxion of x 3 is x X 3 X x 2 = 3 x 2 x ; and the 
fluxion of x 4 is x X 5 X x* 
: 5 X"* x. 
In the 
same manner the fluxion of a -j- y^isTy X 
a -f- y} 6 -, for the quantity a being constant, 
y is the true fluxion of the root a y 
Again, the fluxion of a 2 -j- z 2 ] | will be § x 
2 z z x a 2 -j- z 2 ] J : for here, x being put 
= a 2 -J- z 2 , we have x — 2 z z; and there- 
i 
: » 2 x. 
fore 
for the fluxion of x | (or 
a ‘ -j- z 2 l |) is = 3 z z v' a 2 -f- z 2 . 
3. To find the fluxion of the product of 
several variable quantities, multiply the 
fluxion of each, by the product of the rest 
of the quantities ; and the sum of the pro- 
ducts, thus arising, will be the fluxion 
sought. Thus, the fluxion of x y is x y -J- 
y x ; that of x y z, is x y z -\-y x z z x y ; 
and that of nxyz is vxyz + XVIJZ 4- 
j/V'XZ + zvxy. Again the fluxion of 
a -j- x X b — y = a i -|- 6 x — ay — x y, 
is b x — ay — x y — y x. 
4. To find the fluxion of a fraction, the 
rule is, from the fluxion of the numerator, 
multiplied by the denominator, subtract 
the fluxion of the denominator multiplied 
by the numerator, and divide the remainder 
by the square of the denominator. Thus, 
In the examples hitherto given, each is 
resolved by its own particular rule; but 
in those that follow, the use of two or 
more of the above rules is requisite : thus 
(by rule 2 and 3) the fluxion of x 2 y 2 is 
found to be 2 x 2 yy -j- 2 y 2 x x\ that of 
X ^ 
— , is found (by rule 2 and 4)] to be 
2y 2 xx-2x_ 2 yy ^ ^ rf xfyj 
y z ’ 
(by rule 2, 3, and 4,) found to be 
2 x 2 yy 2 y 2 xx x z — - x 2 y 2 z 
z 2 
5. When the proposed quantity is affect- 
ed by a coefficient, or constant multipli- 
cator, the fluxion found as above must be 
multiplied by that coefficient or multipli- 
cator : thus, the fluxion of 5 x\ is 15 x 2 x ; 
for the fluxion of x’ is 3 x 2 x, which, mul- 
tiplied by 5, gives 15 x 2 x. And, in the 
very same manner, the fluxion of a x n will 
be n a x n_1 x. 
Hence it appears, that whether the root 
be a simple or a compound quantity, the 
fluxion of any power of it is found by thq 
following general Rule : 
Multiply by the index, diminish the in- 
dex by unity, and multiply by the fluxion 
of the root. Thus the fluxion of z 2 — 
8 z 7 z : the fluxion of 4 x 6 — 24 x 5 x and 
. „ • « x . y x — ■' x y ,, , 
the fluxion of is — ■- ■ that 
y y 
x 
x X X -\ - y — X y X 
x 
I \ 2 
x+y; 
12 
Z - zz: — z 
5 20 
z = 
of 
or 1 
* + y ’ x +^] 2 
yA--*! ; and that of *±JL±1, 
^fy\ ! X + V 
z . z X X y — X y X Z 
+ x-|- y’ ,S 
and so of others. 
the fluxion of 
3 ss 
5 z i" 
Having explained the . manner of deter- 
mining the first fluxions of variable quan- 
tities, it is unncessary in a work of this 
kind to enter upon the second, third, &c. 
fluxions, we shall therefore proceed to 
Fluxions, inverse method of, or the man- 
ner of determining the fluents of given 
fluxions. 
If what is already delivered, concerning 
the direct method, be duly considered, 
there will be no great difficulty in con- 
ceiving the reasons of the inverse method : 
though the difficulties that occur in this 
last part, upon another account, are in- 
deed vastly great. It is an easy matter, 
or not impossible at most, to find the 
fluxion of any flow ing quantity whatever ; 
but, in the inverse method, the case is 
quite otherwise ; for, as there is no me- 
thod for deducing the fluent from the 
fluxion a priori, by a direct investigation ; 
so it is impossible to lay down rules for any 
other forms of fluxions, than those particu- 
lar ones, that we know, from the direct me- 
