0 Og Og Cpe x! 
~~ 
FLUXIONS. 
As h denotes here any quantity whatever, ‘we may put Now, when x= 0, then sin. x= 0, and cos. # = 1, 
x instead of h, and then we have - 
- +, 2? ” x3 Vv Ca oy 
fla)=U 4 U'e4D" 5+ UO" 55 + Ua gg + fe. 
This elegant formula for the devel tof a func- 
ila BE of Inte years, Been sacibad by the French 
penne sar estione Commer Irs ae 
rin, who gave it in his Treatise o ; 
Art. 751, (printed in 1742). : 
that Maclaurin seems to consider it as identical with 
Taylor's theorem, to which indeed it ie stvetly piles. 
ce- 
suppose, Ist. that f (x)=u=(a-+2)*, m being any con- 
stant be then, - 
; Tenney, 
ean (n—1) (a+2)"—, 
ten (n—1) (n—2) (af2)"-3, &e. 
when r=0, then u or (a4.)" becomes a” which is 
therefore the value U. Also, by making #=0 in the 
du du du : 
values of > 750 qv &% we set 
U'=na—,U"=n (n—1 a", U" =n (n—1)(n—2) 0", 
&c. These essions, when substituted in the gene- 
tal theorem, us 
(a4+2)"=a"+4+n ama Det = 
HONE sy te 
Thus we haye another demonstration of the binomial 
2d. Next let it be required to develope the functi 
log. —— series. In this case, putting B for 
( Beneneneee Pte) weds 
. u 
u=log. (a+2) ; n= Bays)’ dv Bafa apny? 
Gu 2 du 2.3 
qa=t idea Blapaye * 
of mele Nd A ae 
eaten «ot 2 bedaiaes ; =U’; in like manner we 
= dx Ba 
; 1 at 2 J 238 
find Wa— wu =p "=A & 
Henee, from Dafoe, re a) ci 
log. (a2) = logay + {2 — =. 4 a0} 
the same expression as was found in Art. 53. 
3d. Next, let /(2) = u=a*; then putting A for 
1.(a) oo here —. =Asa*, &c. Now 
)  dacrniyg nT ? dxi ee 
when x=0, then a*=1, therefore, in this case, 
= U=1, U’ =A, iy ge =A}, &e. 
substituting these in the general formula, we get 
et as, Bey Dae - 
ets” as ee 
4th. Lit fiegieen = sin. 2; in this ease * = cos. x, 
dtu ‘ du u J 
Ge =~. = — 008. 7, Gy = Sin. x, &e. 
415° 
Direct 
Method. 
U=0, U'=1, U0, UY =—1, U"=0, & =“ 
and the general formula gives in this case, 
x + 2 2: x? 4 we 
23° 2345 , 2.3.4.5.6.7 “e 
3 2 
5th, Iff( «)=u=cos. 2; then 5 =—sin. 2, = 
Bu dtu 
sin. z= — 
— 00S. 2, 73 = sin. x, Toy = 008. x, &e. When x=0, F 
then cos. 7=1, and sin. c=0, therefore, 
U=1,'U’=0,; U’”’=—1, U0, UVs=I, Bie; 
at x6 
asa zsaset & 
In like manner, we may develope the function 
u= tan. x, and w= sec. x into series; but the expressions 
du d 
for a <=, &c. will be more complex than in these 
and cos, « =1— o+ 
Rxemples. On the other hand, we may find series 
whi 
shall express the arc by means of the cosine, 
the sine, or the tangent, &c. We select the last as the 
most simple. Let /(x)—u= are, of which the tangent 
is «: then tan. u=2, said wit ek (art. yeas 
142% sec.*u 
=d x cos.*u; therefore, 
du m 
Fg 008? He (1) 
Regarding now cos. w as a function of u, which again is 
a function of 2, and considering dz as constant, we haye 
(by rules (A) and (D) art. 26.) 2 =—2dusinu 
X cos, u, but we have seen that du = dx cos.? u, there« 
fore, substituting for du its value in the second mem- 
ber of the last equation, and also putting sin. 2u for 
2 sin, u cos. u, we have 
Pu 
dz —— Cos*wsin.gu. - (2) 
From this 
tion, by taking the fluxi rul 
(F) ast 9, and rules (A), (D) art 26.), ‘we > fa 
u 
da = — 2 4x cos. u (cos. 2 u cos, u—sin. 2 u sin, u). 
_ Let the value of du be substituted instead of it 
in the second member, and also cos. $u instead of 
cos, 2u cos.u—sin. 24 sin. u, to which it is equal, 
(Anrrumeric of Sines, art. 7.), then dividing by dz, 
we have — 
Gu ' 
Tat = — 2 008.3 u cos, 3 u. (3) 
From this equation, again, by proceeding as before, 
we find 
d*u é 
r= 2.3 cos.4 u sin. 4 u, (4) 
du P 
as = 2.3.4 cos.’ w cos. 5,” (5) 
&e. . 
Now when «=0, then u= 0, and cos.u, cos.3u, 
cos. 5u, &c. are each =1; also sin. uv, sin. 2u, sin. 4 u, 
&c. are each =0. Hence, from the formule (1), (2),. 
(3), (4), &¢. we have 
U=0, U1, U’”=0, U”= — 2, UY =0, UY=2'3.4, 
&c. and recurring to the general formula, we get in 
this case : a 
xe 
u(=arc to tan. “= Zt ci 
The preceding investigation affords an elegant ex- 
ample of the utility of the arithmetic of sines in the 
fluxional calculus. 
&e, 
