FLUXIONS. 
48j 
In the direction perpendicular to SC. Put SC^ry, then 
Es—y ; and, by fimilar triangles EC s, CSY, E.s fjj : CE 
:CS (y) : S f— - yxCE . 
Cor. If the point C have no motion in the direction 
SM, the curve deferibed will be a circle, and Es becom¬ 
ing =o, the cotemporary fluxion of a circular arc whofe 
radius SC revolves with the fame angular velocity, will 
be CE. 
32. With any radius SA deferibe the circle ABD, pro¬ 
duce SC to B, and SE to v meeting Bv a tangent to the 
circle ; and fuppofe the angle ASC to vary as SC™. Put 
AS—r, SC—_y, ABr=x, Bv—x cotemporary with the 
fluxions C E, Es; then, as x is the meafure of the angle 
ASC, let us fuppofe that when x becomes any quantity 
s,y becomes t, and x : s : :y' 
z=.x—Bv ; and 
CE; 
r 
S T_ 
and 
msy"— x y 
t" 1 
t m 
msy ”-+ 1 
rt m 
by fimilar 
msy'"y 
rt"‘ 
triangles SBr?, SCE, r:y :: 
±CEj 
hence,by Art.27. SY 
(=* 
Cor. If SZ be perpendicular to CY, we have, by fim- 
ple triangles,YSC, SCZ, CY : CS :: CS : CZ- — • 
Ex. 1. Let the curve be the fpiral of Archimedes. 
Here m— 1, 
sy 2 
and SY= — ; 
rt 
hence, CY — ^J- 
(s 2 v 4 
y\/y 2 +i > 2 
b ’ 
putting 
r 2 t 2 
—b 2 ; therefore CZ: 
-+y = 
by 
'Vy 2J r^ 
Hence alfo, SZ ——^—-. 
Vy 2 +t > 2 _ , 
Let the curve be the reciprocal fpiral. 
Here 
st 
and SY=-, a conftant quantity. 
r 
Let the fpiral be the lituus. Here 
—2 st 2 
Ex. 2 
m— —i, 
Ex. 3 
and SY 
ry 
Ex. 4. Let th.e curve be the logarithmic fpiral. This 
curve is generated by the uniform angular motion of SC 
about S, whim C recedes from S with a velocity propor¬ 
tional to SC; hence, sE, the fluxion of SC, varies as SC; 
but as the angle CSE is always the fame in the fame 
time, CE will vary as CS ; hence, CE : Es :: a : 1, acon- 
n • CE a cv .^XCE r 
ftant ratio, .•.-jr-=a, and SY=——— —ay; conlequent- 
ly SY : SC :: ay : y :: a : 1, a conftant ratio; hence, the 
triangle SCY continues always fimilar to itfelf, and there¬ 
fore the angle SCY is conftant, and is known from the 
ratio of a : 1. 
Of the BINOMIAL THEOREM. 
Prof. XIII.— To exprefs the value of o±a)" by a feries. 
33. The fquare of i-j-x is i+2x + x 2 \ the cube is 
i-E3x4-3a: 2 -1-x 3 ; &c. hence it appears, that the coeffi¬ 
cients do not depend upon the value of at, but upon 
the index of the power ; therefore if x be diminiffied 
and at laft vanifh, it will make no alteration in the co- 
Vol.VII. No.443. 
efficients. And as by the continual multiplication of 
we maniteftly get a quantity with all the powers 
of x regularly afeending, let us affume i-fxV'=i fax 
-\-bx 2 -\-cx 3 - > r dxf-' r &c. Now to determine the values 
of a, b, c, d, &cc. take the fluxion of both fides of this 
equation, omitting x as it will be common to every term ; 
then take the fluxion of the refultitig equation, and lo on 
continually, and we get the following equations. 
?2 X i +x|”— 1 2=a-f-2ix+3«: 2 -i-4'fx 3 -{- Sec, 
77 .a —1 x i+.vl — 2 — 2 b-+ 2 .' 7 ,cx+- 7 ,.i r dx 2 +- Sec. 
n.n — i.n —2 x i-j-xY'— 3 22=2.3c4-2.3.4rfA-f- &c. 
Sec. Sec. 
Now make x=.o, and from the firfl equation, 77222 a; 
from the fecond, n.n — i—zb\ from the third n.n — i.v —2 
=2 .3c, eec. hence, a—n : b—n. - ; c—n. -.-> 
2 23 
Sec. where the law of continuation is manifeft. Hence* 
. ■ 72 I 71 j u 71 2 
1 +21 ’'=22i-{-?7A'-J-?z.- x 2 +-n. -- .-x 3 -h- &c. Now 
2 23 
if n be a whole pofitive number, it is manifeft that this 
feries will terminate, for we muft cotne to the coefficient 
n — 1 v—n 
n. - .... -—o. But the above inveftigation holds, 
2 , ??-fi fa 
whether n be a whole number or fraftion, pofitive or nega¬ 
tive. If«be a negative whole number, the feries will 
never terminate, becaufe the numerators??, ?z—1, n —2, 
&c. become then —??, — 71 —i, —?2—2, 8 cc. and therefore 
can never become —o. Alio, if s be a fraftion, it is 
manifeft that 11, n —1, n —2, &rc. can never become 2=0, 
becaufe a fraftion can never be deftroyed by the fub- 
traftion of a whole number from it. Hence, the feries 
will always run on ad infinitum, unlefs n be a whole pofi¬ 
tive number. If the binomial be 1— x, then a: becoming 
negative, the odd powers of x will be negative and the 
even powers will be pofitive; hence, 1 —a] 1=21— nx+- 
v- 
71 - 1 
n - -x 2 
2 
- 71 .- 
-i n- 
-x 
3 1 
Sec, 
34. H ence, we may expand a+x |\ For 
®+2=«Xi+-, . ‘.a -f- a )’‘=za ’ : X 1 +^j — ( by writing 
for x in the feries in the laft article) a" y. 
X 71 - 1 
1+n. —\-n . - 
CL 2 
V— 
71 - 
v 3 
-1 . 
1 a- + n.- 
-a”- 
2 x 2 -\-n . 
. — + Sec. —a"+. 
a 3 
~ 3 * 3 + Sec. 
2 23 
For the different cafes where the feries converges or 
diverges, or becomes 2220, fee Dr. Waring’s Med. Anal. 
P- 4 ! 5 - 
The principal ufe of this rule is to extraft the roots of 
binomials; for if k be a fraftion, the feries gives that 
root of the binomial which the fraftion exprelfes. 
Ex. j . What is the fquare root of a 2 +z or the value of 
a 2 +zd\\ in a feries ? By Wood’s Elements of Algebra, 
Art. 250. a 2 + z 2 \%: 
z 2 
- <2 X 1 + — 
a 
2 
compare 1+—“with 
1 +.*(", and we have — 222X, -2 —n ; hence, by fubftitution, 
«X 1 +— 
a ° 
&C. =22fl-|---- 
2 a 8<2 3 
1 z- 1 z 
20 XI- 
2 a 2 2 
■ r A 2® 
+ " 
• + ; 
160 5 
Sec. 
Ex. 2. What is the fourth root of 1— x. 
■ + 
or the value 
1 
I 
of 1 —aP in a feries ? Here 72=2:-, and 1—*r=i — ‘ x + 
4 4 ' 
1 a — 1 , i 1 i 2 . <?- 1 1 .. 3 
4 2 
3-7 
4.8.12 
4 2 
a: 2 —, &c. 
-2 - , „ 1 
-a 3 +, Sec.2=1— -x—--x 2 ' 
3 4 44 
6 H 
