7 PLUTON. 
Disest —fluxion of u (denoted by du) will be the fluxional co- 
ee, Soin of 
Method. 
ym Fe that is by d2*;* and, on 
+ 
nth order gig og by the nth power of 
e conta like as the 
fiuxional coefficient of the first order is J", the ux. 
ional coefficient of the mth order will be St 
43. It has been proved, (art. 28.) that if x and» are 
two functions, which are equal for every value of x, 
then the fluxional coefficients of thei order derived 
u dv 
from them will be equal, that is, dendet Now, as 
these expressions denote other functions of x, having the 
same as the original functions, we must, jn like 
i, q 4) ¢ (42) . 
dz dz au dy 
manner, have az = 
so that not only will the fluxional coefficients of the first 
order be equal, but likewise those of the second and all 
i orders. 
As the equations } 
a du__dv au d*v & 
Sas S Fo Ge = an? *% 
may also be expressed thus, 
d(u—v) d:(u—v) 
u—v=0, “Tz =O dat a 0; &ce. 
it appears that the same proposition ay yr pars 
sed as follows: If X be such a function of x, that =0, 
for all values of x whatever, then shall ?* — 0, —— 
ox dx dx 
= 0, and universally——— = 0, , 
> 2a 1 1 
For example, if X r r ee ae 
pression which is always — 3 + 
dz (@—ay * (z—a)— z 5=°% 
fe. opie (= 2 ) ( ett a] 
dv ‘(2*—a*)3 Go ~(z y Bye 03 Ke. 
44. The rules for finding the first fluxion of a func. 
tion of 2, are equally applicable to the second and high- 
uxions. 
Let u—a" ; then (rule (A) )du=n 21d 2. By consi- 
dering dx as a ee Fae: we have d(du), 
that is, d*u=n(n—1) 2—?d 2?, 'y proceeding in this 
manner, the successive fluxions of u—z" are 
; du=n xd x, : 
d* u=n(n—1)a*—2d x2, 
d? u=n(n—1) (pricey &e. 
‘From these expressions, that when n_is 
whole number, the flaxion of the ath hier, b-m te 
stant quantity, and therefore all the following fluxions 
vanish. 
log. x ; then 4, i the basis of the sys. 
Let u= 
tem, by rules (B) and (A), we get 
to ees aiid at __ 2d 23 
du= a Ms #1 Oyo "= aig °& 
Let u=e*, ¢ being the basis of the Napierian system 
of logarithms ; then, by rule (C), bd 
~ duxdae’, dtu=date’, du dae" -&e. 
VOL, Ix. PART IL. s 
de >thatis, Fa = Go . 
409 
From these examples, it appears, that the functions ‘Direct 
aS 3 Method. 
w=log. x, and u=e have fluxions of all orders whatever, Swe 
This_will also be found to:be true of the functions 
“= sin. x, and u= cos. x. 
Of Fluxional Equations. 
45. We have hitherto supposed ‘that the expression 
whose fluxional coefficient is. to -be determined, was an 
explicit function of the variable quantity 2, that is, a 
function of x of some given form. But it may be res 
quired to find the fluxional coefficient of y, an implicit 
function of 2, the nature of which is expressed by an 
equation, For example, the relation of y to x may be 
expressed by the equation 
if y2 me y+2*—a = 0. 
In this particular case, by resolving thg equation, we 
have 
Y=MTH Vf J ar+(m—1)x* b ; 
as y is now an explicit function of z, its fluxion may be. 
found by the rules already given. But the equation 
which expresses the relation of y tox may not admit of 
being resolved ; and when this is the ease, the fluxional 
coefficient must be determined upon principles which. 
we are now to explain. 
As we have denoted any expression of calculation 
composed of x and constant quantities by the symbol 
J(#), we may, in like manner, denote any expression. 
Bobet ofe, 
si y, and known by by S(=, 9). 
n this way, an uation, such as 2 mry+ar— 
a=0, aipeeelag the relation puewenl cant nney be 
briefly indicated thus, F(«,y)=0. * Now, al ugh we 
should not be able to resolve the equation, we may be 
certain that y is expressible in some way or other by 2: 
It may therefore be assumed that y=X, where X de- 
notes some expression of calculation made up of x and 
known quantities. This value of y being put instead 
of it in e equation F(x, y)=0, it becomes (a, X)=0, 
an equation involving only x and constant quantities : 
And as the equation F(x, y)=0 holds true for every 
ible value of x, so also must the equation F(«2,X)—0: 
is must therefore be an identical equation, and con- 
sequently it will have the ies which (in art. 43.) 
have been proved to belong to such an ion: So 
that putting u to denote briefly the expression F(x, X), 
or its equivalent F(2,y), as we have u=0, we must 
du ui. du 
have also —-= 0, —— 
da dx* 
which mean that if the fluxion of usF (2,4), (consi 
divided by dz, 
46. Let y be a function of x, of such a nature that 
¥+e=0%, or 
¥+2—a=0, 1) 
a being a constant quantity. _ In this case, 
uy +2°—a’*, 
therefore, taking the fluxions and dividing by dx, we 
