Direet 
Method. 
—o 
408 
sin, r= 2 sin, 3 x cos. 4 r cos. 3.2; 
because sin. } =sin. 4 « x 2 cos. 4.x; therefore 
sin. r= 2 sin. 4 x cos. $ x cos. } x cos. 4 x. 
And, in general, putting m for of cosines, 
X cos. i x, 
and 2=2", 
sin. en sin. } zcos.f2cos.iz. . 
Hence, proceeding as in Art. 37. we have 
d.sin.x__nd.sin. tx d.cos.42 d. cos. 3.x 
sin.s ” =6ansin. Se" cond” * * cos. iz’ 
actually taking the fluxions of the numerators, 
snd rjocnieg the common factor dx, we have 
cos. FCO, Fe sin. 4x sin.} x 
tin. « ~ nsin.ie 200s. } x 4cos.42 
and hence, transposing and putting tan. x for at &c. 
1 
: ere ie eee 
ntan. is ~ 
+ftan.42.....42 tan. ty 
This formula is true, whatever be the number of 
terms in the series. Let us now suppose the arc x 
indefinitely small, and n indefinitely great ; in that 
n tan, } x=are 2; hence, supposing the series to 
cn ad iefaitom, we have 4 
1 i 
Stee Ft det pan. fa F tan F2+&e, 
This expression for the reciprocal of an arc, 
Bh cote found in Arrrumetic of Sines, Art. 
sg nr ae maar naa owt “et a 
By taking uxions repea oO sides of this 
equation, we may find veries for reciprocal of the 
squares, and any higher powers of an arc. 
40. From the few applications which have been given 
in this Section, it must appear, that the fluxional calculus 
is F peewee Yee in analytical inquiries. For 
as all quantities whatever may be treated by the ele- 
mentary ions of addition, subtraction, ipli 
FLUXIONS. 
iginal function, 
finding the 
fluxion of a function. For example, if u = x*, then by 
rule(A), 4 = nx*-!, therefore in this case p=n x”, 
and since dp=n(n—1) 2*-*d x, rule (A,) there- 
fore q = SP=n (n—1) a2, In like manner, dg= 
by the rules which have been given 
n(n—1) (n—2)x-5dx, and hence r atts 
dx 
n(n—1)(n—2)2"—5, and so on. : 
The relation that each of the fluxional coefficients p, 
q, 7, &c. stands in to the original function 1, is indica- 
ted by heap the fluxi coefficient of the first or- 
der, q that the second order, ¢ thasof the third order, 
and soon; so that in the case of the function u=2", 
the fluxional coefficients of the first, second, and third 
order, mis qtetetiyene tact os ( we 
nal, g=n(n—1 2, r=n(n—1) (n—2 3. 
"2. The jroo by which the luxiondl eneticientsp, 
y, 7, &e. are to be determined from the function w, is 
du d dq 
‘\ bon 5 I= 7 hae Fs &e. 
In these, the conventional symbol dx enters merely 
faction | erin, iat pertrasing’ the operons, ve 
; , in i operations, we 
= treat it as if it represented some constant quantity, 
and so we shallhave - 
eae eT ARLE a 
erate PO 
cation, and division, by attributing to them the proper- 
ty of being SENSE Stree ert and, therefore, aSa(du) ‘a 
ration, namely, that by which their uxions are taken. _du __—d(du) AS —? 
It must also be obvious, that the use of the character P=gz 9=qy-de’" = “dardacda > but instead 
dx is merely to shew, that the changes of magnitude of 
different functions are all referred to that of the variable 
quantity x, which, in a function of a determinate form, 
has the same relation to the function that a root has to 
its power. ; 
Of the Different Orders of Fluzions. 
41. In what has been already explained, we have es- 
tabeioeed (Oe Mppoctent peipeiple in. shel sis, that if u, 
any expression of calculation, be 
susceptible 3 
there is a certain p, deducible from u by de- 
terminate rules, which is a limit'to the ratio of the cor- 
eS ee and which we have 
the fluxional co-efficieni, originating from the 
uu : 
Now, by applying the same hypothesis to p, that is, 
of repeating the letter d so often, it will be better 
to put d*u for d(du), and d3u for d }d(du) + ,and, 
and hence again, 
<aenes, du=qda*, Bu=rd x, &e.* 
hepa ache end das the peed fcies ot 
rst 10n 9 u, 
ml do te third fuzion, and 80 ! 
* The reader must be careful to give each of these three symbols, d*u, dw* d(w) its true meaning. The first indicates 
ration of finding the fluxion of u is to be performed n times; the second, that the result of the first operation is to be raised to the nth 
power ; the third, that the fluxion of the nth power of w is to be taken. 
4 
ON 
