Direct 
: Method. 
= 4 
“FLUXIONS. 
substituting 2h for 2, the function becomes (x-+-h)”. 
Now, 
r+hyao"+n2*—ht+Hh, (Art. 19. 
Mitnch oh be: neaty ctnot asc Laut 
consists of the first power of h multiplied by a coeffi- 
cient which is independent of h; therefore, in this case, 
p, the fluxional coefficient, is n 2"— ; and changing / in~ 
to dx, we have 
Hence this rule, 
du=nazt— dx, 
pee Et eam : 
The fluxion of 2*, any constant ‘power of a variable: 
quantity, is the continual ct of the ent; a 
power of the same quantity, whose index is one less 
Lespob index of the power proposed ; and the fluxion 
tity. 
IL. pate x a : 
log. (2-+-h)=log. r+ zr@jth h; (Att, 19.) 
Here 6 denotes the basis of ce, and 1. (4), the 
Napierean logarithm of 4. Therefore, in this case, the 
fluxional co-efficient is, and 
«1, (6) 
dat dz 
_, £h (oy 
Hence this rule, ss 
The fluxion of the lo G of a variable number x 
is a fraction whose numerator is the fluxion of the num- 
ber, and denominator is the roduct of the number by 
the Napierean ithm of the basis of the system. 
Notr.—In the may 
common system we have found, (Art. 12.), 
- 1 (6) =2.802585092094= , 
uti 
III. Let the function bew=a", a being constant, and 
xvariable. We have found that 
bh 
system, l.(4)=1. Inthe. 
aw" =a fail. Hh, (ar 
Thievefne;"in this base; Whe titcionel Coatboont pox 
a 1. (a), and 
du=a'drl.(a). 
by pire on eh rr Be 
The fluxion of a variable power of a constant quan- 
tity is the continual ct of that power; the Na- 
logarithm of the constant quantity; and the 
of the variable 
IV. Next, letu=sin. r,u=cos.c, then we have (art. 19.) 
Sin, (z+A)= sin. rh cos. c+Hh; 
Cos. (x ates bach ith 
In the first of these ions, the fluxional co-ef- 
ficient is cos. z, and in the second it is — sin. x; there- 
fore, dv=d (sin, 2)=d z cos. x ; 
du=d (cos. c)=—d z sin. x. 
of the sine or cosine of af arc to that of the arc 
(article 17.), we have considered the arc as the 
ma basis, Mean snd conta te Sy Toriction. 
it to see expression for the ratios, that 
the lintit wil! be the very same, if on the contrary we 
consider the sine or the cosine as the variable basis, and 
the are as the function. Since, then, upon» the ‘first 
In 
ments 
hypothesis, we have the fluxional ratios “(2-#) — 
cos, x, and 
==) = —sin.2; we have, upon the 
determining the limit of the ratio of the incre-’ 
other hypothesis, the fluxional ratios, 
ey ROOERE TepEE Urs i 
d(sin.z)~ cos.x’ d(cos.z)~ sin.” 
Or, putting u instead of x, reserving x to denote the 
variable basis of the function, 
be re du 1 
d(sin.u) ~ cos.u’ d(cos.u) sin. 
Now put sin. w=z, then cos. u=4/(1—«*), and from 
the first of these formule we have; - 
du te) dx. 
dr 1=2*)’ and inte V4 Gath 
Next, let cos. w=2’, then sin. w=4/(1—.2*,) and: 
from the second formula we have: 
Ss ee Re sold gf tag tt Lod 
da’ ~ /(i—27y = I—2)" 
Hence the fo wing rule; which applies alike, whe- 
ther the sine and cosine be considered ‘as functions of 
the are; or the are beconsidered as a function of the: 
sine or of the:cosine.: —. 
_ (D) 
The fluxion of the sine is 
are multiplied by the cosine: and the fluxion of the 
are is equal to the fluxion of ‘the sine divided by the 
cosine 
the arc multiplied by the sine, and the negative sign 
prefixed to the result; and the fluxion of the arc is equal 
to the fluxion of the cosine (with the sign—) divided * 
by the sine. ; 
27. Let v and x be any functions of «, and let it be 
p to-find the fluxion of y=a+d 
a, b,c, denote constant quantities, 
Let p and p’ be the fluxional coefficients of v and « 
equal to the fluxion of the.- 
v—c u, where - 
The fluxion’of the’ cosine ~is equal to the:fluxion. of 
respectively. Then, (art. 19.) when # becomesix-h, . 
and u bécomes wp! , 
fooere 
us su when x 
% x Hi h+Hh) 
=a —c(u-p’ 4 
ih =e har ipmep (oot as 
that is Lat (pep he ; "h, 
Hetheoh ieee dg as iabecab bsniatolonbleg 
ence iy iS 
therefore dy=(6 a 
of & 
of vanishing when h—0: 
ut p and p’ being the fluxional co-efficients of y and 1, 
pdz=do, and p'dz=du’; therefore 
dy=adv—bdu.. . 
Hence this rule,, 
E . 
The fluxion of a Cones nate consists of several 
terms, is the sum of the fluxions of theterms, each re- 
taining the sign and co-efficient of the term. 
ekg one fluxion of a constant term:is to be reck- 
oned =0. 
28. It ma appear almost self-evident, that if 
two functions of t hee variable quantity be-equal, their 
fluxions will also be equal. We shall, however, demon- 
strate this proposition, because of its great importance 
in the theory. 
Let the two functions be f(z) and F(x), which we 
suppose are always equal, ‘although they may: be of a 
different form : For example, f(r) may be =~, and 
1 
it , two expressions which form 
14x 
ah, then y be- 
ip—ep’ )dx=bpdx—cp'da': - 
