banner the fluxions of second fluxions arc call- 
ed third fluxions, and denoted by the same let- 
J. . t 
ters with three dots over them, as x, y ; and so 
on for fourth, fifth, &c. fluxions. 
The whole doctrine of fluxions consists in 
solving the two following problems, viz. 1 . From 
the fluent, or variable flowing quantity given, 
to find the fluxion ; -which constitutes what is 
called the direct method of fluxions. 2. From 
the fluxion given, to find the fluent, or flowing 
quantity ; which makes the inverse method of 
fluxions. 
Direct Method of Fluxions . — The doctrine of 
this part of fluxions is comprized in these rules : 
1. To find the fluxion of any simple variable 
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 compound quantity x 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 expo- 
nent. This rule is expressed more briefly, in 
algebraical characters, by nx x — the flux- 
ion of x'\ Thus, the fluxion of a-’ is x X 3 X x 2 
= 3x 2 x ; and the fluxion of a- 1 ' is x X 5 X * 4 
= 5v 4 a-. In the same manner the fluxion of 
a -j-3.1 7 is 7 y X ; for the quantity a 
being constant, y is the true fluxion of the root 
1' 
Again, the fluxion of a 2 -f- 
X S-\ -z 2 Y 
will 
« 4-> 
be A X 2z« X " 4 + z 7 ) 1 : for here, a- being 
put = a 2 -\- z 2 , we have a z= 2 z»; and there- 
fore A 
C-, for the fluxion of xA (or a 2 z 2 ) " 2 ") 
is ~ 3zss if a 2 4- z 2 . 
3. To find the fluxion of the product of se- 
veral variable quantities, multiply the fluxion of 
each, by the product of the rest of the quanti- 
ties; and the sum of the products, thus arising, 
will he the fluxion sought. Thus, the fluxion of 
Ay is A y 4->- ; that of xyz, is xyz yxz -}- »a y; 
and that of vxyz, is vxyz -j- xvyz -j- yvxz -f- 
ievxy. Again, the fluxion of u -|- x X b — y — 
ah -j- bx — ay — Ay, is bx — ay — xy — yx. 
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 deno- 
x yx — xy 
minator. Thus the fluxion of — , is ; 
y y 
that of 
. x X x -f- y — x -\ - y X x 
is - - — — — 
*+y x + yl 2 
’± and that of A ‘ + 
+ > 
or 1 -J- 
as X x -j- y — x -j- y X 
; and so of 
x -r y x -}- y 
others. 
In the examples hitherto given, each is re- 
solved by its own particular rule : but in those 
that follow, the use of two or more of the above 
rules is requisite : thus (by rules 2. and 3.) the 
fluxion of a- 2 y 2 is found to be 2x 2 yy -J- 2y 2 xx ; 
that of — 2 , is found (by rules 2. and 4.) to 
y 
be 
2y 2 xx 
y.y . 
and that of — — , is (by 
rules 2. 3. and 4.) found to be 
2x 2 yy -jr 2y 2 xx X z — x 2 y 2 ii 
FLUXIONS. 
5. When the proposed quantity is affected by 
a co-efficient, or constant multiplicator, the 
fluxion found as above must be multiplied by 
that eo-efficient or multiplicator: thus, the 
fluxion of 5-v 3 , is 15x 2 x ; for the fluxion of x 1 is 
3x 2 x, which, multiplied by 5, gives 1 5x 2 x. And, 
in the very same manner, the fluxion of ax" will 
1 " ~ 1 • 
ue nax a*. 
Having thus explained the manner of deter- 
mining the first fluxions of variable quantities, it 
remains to say something of second, third, &c. 
fluxions. We have already observed, that the 
second fluxion of a quantity is the fluxion of the 
first fluxion ; and by the third fluxion is meant 
the fluxion of the second ; the fourth, of the 
third ; and so on. The fluxions, therefore, of 
every order, are only the measures of the velo- 
cities by which their respective flowing quanti- 
ties, viz. t-he fluxions of the immediately pre- 
ceding order, are generated. Hence it appears, 
that a second fluxion always shews the rate of 
the increase or decrease of the first fluxion ; and 
that the third, fourth, &c. fluxions differ in no- 
thing, except their order and notation, from first 
fluxions 5 and therefore, are also determinable in 
the very same manner, by the rules already laid 
down : thus (by rule 4.) the (first) fluxion of x 3 
is 3x 2 x ; and if x is supposed constant, that is, if 
the root x be generated with an equable or uni- 
form velocity, the fluxion of 3x 2 x (or 3x X x 2 ) 
again taken (by the same rule) will he 3 v X 2xx, 
or 6a- v 2 ; which, therefore, is the second fluxion 
of x\ Again, the third fluxion of a 3 , or the 
fluxion of 6xx 2 , is found to be 6x 3 ; further than 
which we cannot go in this case, because the 
last fluxion, 6a- \ is here a constant quantity. 
In the preceding example, the root x is sup- 
posed to be generated with an equable velocity: 
but if the velocity be an increasing or decreas- 
ing one, then x, expressing the measure thereof, 
being variable, will also have its fluxion, which 
is denoted, as said above, by x ; and the fluxion 
of x by x, and so oh with respect to the higher 
orders. 
Here follow some examples, in which the 
root x (or y ) is supposed to be generated with 
a variable velocity. Thus, the fluxion of x 3 
being 3x 2 x (or 3x 2 x x), the fluxion of 3x 2 x x, 
considered as a rectangle, will (by rule 3.) be 
found to be 6xx X x -J- 3x 2 x x — 6xx 2 -f- 
3x 2 x ; which is the second fluxion of x 3 . More- 
over, from the fluxion last found, we shall in 
like manner get 6x X x 2 -f- 6x x 2xx -j- 6xx x x 
-j-3.v 2 X x (or 6x 3 -}- 18xxx -f- 3x 2 x) for the third 
n 1 
fluxion of x 3 . Thus also, if y — nx i-,then 
will y — n X n — 1 X -v 
« — 2 . 
n — 1 
and if 2s 2 — xy, then will 2isz == xy -j-jyx : and 
so of others. 
The reader is here desired, once for all, to 
take particular notice, that the fluxions of all 
kinds and orders whatever, are contemporane- 
ous, or such as may be generated together, with 
their respective velocities, in one and the same 
time. 
Inverse Method of Fluxions, Or the manner 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 conceiving the reasons of 
the inverse method : though the difficulties that 
occur in this last part, upon another account, 
are indeed vastly greater. It is an easy matter, or 
not impossible at most, to find the fluxion of 
any flowing quantity whatever ; but, in the in- 
verse method, the case is quite otherwise ; for, 
as there is no method for deducing the fluent 
from the fluxion a priori , bv a direct investiga- 
5 C 2 
755 
tirni ; art it is impossible tb lay down rules for 
any other forms of fluxions, than those parti- 
cular ones that we know, from the direct me- 
thod, belong to such kinds of flowing quanti- 
ties ; thus, for example, the fluent of 2xx is 
known to be x 2 ; because, by the direct method, 
the fluxion of x 2 is found to be 2xx : hut the 
fluent of yi- is unknown, since no expression has 
been discovered that produces yx for its fluxion. 
Be this as it will, the following rules are those 
used by the best mathematicians, for finding the 
fluents of given fluxions. 
1. To find the fluent of any simple fluxion, 
you need only write the letters without the dots 
over them : thus, the fluent of x is a-, and that 
of ax + by, is ax -j- by. 
2. To assign the fluent of any power of a vari- 
able quantity, multiplied by the fluxion of the 
root ; first divide by the fluxion of the root, add 
unity to the exponent of the power, and divide 
by the exponent so increased ; for, dividing the 
fluxion nx x by dr, it becomes nx ; 
and adding I to the exponent (n — 1) we have 
which, divided by n, gives x", the true flu- 
ent of nx‘ *x. Hence, by the same rule, the 
fluent of 3x 2 x will be — x ! ; that of 2x'x = 
— ; that Of y \y = |jy ^ ; that of ay 5. y __ 
+ 1 
+ « 
3ay • 
; and that of y y 
ny 
+ 1 
that of — , or axx 
1 — *’ 
that of 
. _£+> 
*, and that of a + a 
X 
m — 1 
+ : 
n) n -j- I 
X n 1 
In assigning the fluents ef given fluxions, it 
ought to be considered, whether the flowing 
quantity, found as above, requires the addition 
or subtraction of some constant quantity, to 
render it complete : thus, far instance, the flu- 
n — 1 
ent of nx x may be either represented by 
x* or by x* ^ a ; for a being a constant quan- 
tity, the fluxion of x” ^ a, as well as of x”, is 
n — I , 
11 x x. 
Hence it appears, that the Variable part of a 
fluent only can be assigned by the common me- 
thod, the constant part being only assignable 
from the particular nature of the problem. Now 
to do this, the best way is to consider how much 
the variable part of the fluent first found, dif- 
fers from the truth, when the quantity which 
the whole fluent ought; to express is equal to 
nothing ; then that difference, added to, or sub- 
tracted from, the said variable part, as occasion 
requires, will give the fluent truly corrected. 
To make this plainer by an example or two, let 
y — a-j-r) 3 x x. Here we first find y 
a -j- x V 
; but when y = 0, then 
be- 
; since x, by hypothesis, is then. 
a -j- x ') 4 
z= 0: therefore — — always exceeds y by 
— ; and so the fluent, properly corrected, will 
