402 
Fandemen- convenience which the'party spirit engendered by thedis- 
t. Newtonat different times 
pute has rendered 
of notation ; but that which a 
in 
sed by the character z, that of « by w, and so on. 
Leibnits and his adherents, on the other hand, having 
given the name differential to what Newton called a 
fluxion, they chose to denote ae 3 yore: Seer 
d (the first letter of the word) to the letter indicating 
the variable quantity : Thus what Newton expressed by 
xz and w, they by dz and du. 
In the first ications of this theory, it seems to 
have been a matter of indifference which of the two no- 
tations was employed. However, in the more extend- 
a : she adlenbe, the forsh ‘ sp: 
to have been found the most convenient ; for 
writers abroad, and icularly the celebrated Eu- 
ler, who has improved this theory more than any other 
individual, have adopted it. The almost complete ex- 
tinction of the dispute concerning the first invention of 
ar jices and nore pertounty-Gjousattable disco. 
the subject ; more i i 
veries ae Euler, Lagrange, Legendre, La Place, and 
others, which have been consigned to posterity in the 
of the foreign notation, and not yet expressed 
in of Britain, have rendered the foreign notation 
quite familiar to mathematical readers in all countries ; 
and in this country they seem to have produced a dis- 
position in some to it in preference to that hitherto 
employed. We are even di to think, that the in- 
commodiousness.of the. British notation is the main rea- 
son, why so few improvements in this branch of mathe- 
matical science have been made in the rn 
Viewing the matter thus, we do’not scruple to adopt 
-the foreign notation, as sriplayed by the latest and best 
writers on this subj e accordingly shall denote 
the fluxions of x and « by dz and du respectively ; and 
it must be carefully observed, that the letter d prefixed 
to a quantity, does not mean a product of which d is a 
factor: the letter is prefixed to the quantity to indi- 
cate ce heey cee asi oon letter'r, which has-since de- 
generated into ical sign, was originally prefixed 
to : quantity to open tte tartaze ae ”. 
n expressin e powers of dz; the fluxion of'a quan- 
tity z, we shall denote its pecan power byt: th third 
by d23, and, in general, its nth power by dz. 
By such an expression asd(a+xr-4-cz*), is meant the 
fluxion of the compound expression a+-bx+-¢ 2°. 
The fluxion of the sine of an are may be 
thas, d(sin. x) ; and'the'same notation may be follow- 
ed with to the curve, tangent, &¢. Similarly 
the fluxion of f(x), any function of a variable quantity 
,may beindicated thus, d { /(<). 
24. The theory of fluxions resolves itself into two 
principal ia TR ie ions , 
First, Having giveti the relation of two variable quan- 
tities to find the relation oftheir fluxions. This branch 
hes been called by the Evglich writers, the Direct’ Me- 
thod of Flucions, and by foreigners The Differ ential Cal- 
culus. 
Secondly, Having given the relation ‘of the fluxions 
of two variable 4 ies, te determine thence the rela- 
tion “of the variable quantities themselves. “This: has 
been called in Britain the Jnverse Method, and abroad 
FLUXIONS.” 
the Inte; Caleulus. We proceed to treat of these in 
eh i Z 
SECTION II. 
Or tuz Dinetr Merion oF Fiuxions. 
25. In the direct method of fluxions, the first object 
of inquiry is, how to find the fluxion of any 
function f (2) relatively to the variable q *, 
which is regarded as its basis. The solution indicated 
Te et coool: aliply 8 9p the oAilaty chm 
t i ient; multiply it byt i ’ 
bol dx, which denotes the fy -wr ofthe variable wi. 
ty x ; and the product will be the fluxion sought. Hence, 
e whole difficulty rests upon finding the fluxional co- 
We have investigated the fluxional coefficient in five 
different eases, which comprehend all the elementary 
functions. of a general form, usually admitted into ana- 
lysis. As we wish to investigate the binomial theorem 
and the series for i also the series for sines and 
cosines, by the theory of fluxions, we could not legiti- 
matelyavail ourselves of their aid in establishing its prin- 
ciples, and epee we —s na want of them 
by analytical artifices. In particular algebraic func- 
dioaa;however, the fluxional coefficient is easily found. 
Let us take, for example, the function 23: By substi- 
tating x-/ for x, it becomes 2343 2* h + Srh?+ hi. 
Therefore ("= — 3044 (S2th)h. By com- 
paring this with the general formula-=+") —S* _ 
p+H, we find H=(32+/) h, and the fluxional co- 
efficient. p=3 2°. Hence, fluxion of 2} is 327d x. 
In the case of other integer powers of x, the fluxional 
coefficient and the fluxion are found with equal facili- 
. Thus we have, in like manner, 
ed ll 4 (Gr44chphh; 
and hence, in the case of the function z*, we have H= 
(62°44 2h+4h*)h, and the fluxional coefficient p= 
4.3; and therefore d (x*)=4 23 dz. i 
In these two examples, we have determined not on- 
, the fluxional coefficient, but also H, that part of 
‘No s we do not want ; all ‘we have occasion for, is 
that part of the ratio which is independent of h. 
et i eee ee aaa ae By form, 
(Rf hat -eth+ Ce pach+h) i, 
and the general formula, Ads dl clio 8 un- 
blag Piatra ae 
r+hy=f (x 3 
and it will appear that the 2 soi Gnuling the fluxional 
coefficient may be expressed thus : 
_ Substitute x 4-h in the function instead of x ; 
Baler gf 1, vow valve which is the 5 
and p, a 
and Cos. 2. 
I. Let u=z", being any constant quantity. By 
3 
the" topression for the ratio which vanishes when A=0. 
a 
Direct 
Meshod, 
OL Ee 
ee — 
