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APPLICATION OF THE
INVERSE METHOD OF

FLUXIONS.

The applications of the inverse method of fluxions extend to all parts of the mathematics. But at present we shall confine ourselves to such as are purely geometrical, and which serve as a foundation to all the rest. With this view we shall proceed to investigate formulas for determining the quadrature and rectification of curves, the solidity of bodies in general, as also the formulas peculiar to solids of revolution and their surfaces; and we shall finish with a few examples on the inverse method of tangents.

ON THE QUADRATURE OF CURVES.

107. Let AM represent any curve, and let its axis be AP; and suppose PM an ordinate to the point M; to find the quadrature of the space AMP, draw another ordi- 8 nate mp, and the line Mr parallel to Pp; we

2 shall then have the surface of the space Mmp P=MPx Pp+Mmr. Conceive now that the point m approaches towards the point M, it is obvious that the triangle Mmr will diminish more and more, but it will not

A become zero till the point m falls upon

PP the point M; then M p P will become the fluxion of the space AMP; Pp will be i, and we shall have flur (AMP)=yi, and consequently AMP=syö+C, which by Art. 74

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Therefore also the space AQM = Sxy=C+xyy

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108. Example 1. Let the ci cular arc QMB be a quadrant, described from the

2 centre A, and with the radius a; shall have y=ivaa-ox, and the space

M AQMP = fx Vaa - 2x + C = C + air

– C 23 1.35 1.3 x? 1.3.5 29 2.3a 2.4.5 a3 2.4.6.7 as

2.4.6.8 9a" 1.3.5.7 x" &c.

А

P B 2.4.6.8.10' 11 a? Make x=0, and we shall have AQMP=0, and consequently C=0.

23 1 Therefore AQMP-ar-1.

X5 1.3 27

&c.
3a 2.4 5 a 2.4.6 7 as
16

6 Ex. II. In the ellipse y = v(aaxx). Therefore Syz =

1 (ar

les
.)

-&c.)
3a 2.4 5a3
Ex. III. In the parabola yi=p*..,
and fy:=$p x=fty, or the space

= % APM is two-thirds of the circumscribed ? rectangle. The equation which comprehends 2

M parabolas of all degrees is y="a"-"; therefore, mly = nix + (mn) la, consequently my

A PP or min:: yx: xy :: Syü : fry :: AMP : AMQ. From which it follows that the space AMP is to the circumscribed rectangle APMQ :: m; m+n. Ex. IV. In the equilateral hyperbola

aax xy=aa, and yz = Consequently

N

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Ex. V In the cissoid

✓ax -xx) and y:=x ; (a—x)}; consequently fyr, or the space AKMPA = saiz (a—) Now së (ar-xx)} = the semi-segment AONP; and by art (80) we shall find that fat is (ax)*=* ** (a—x)++ } Sitz' A (a-x)+ Hence ft 3 (0-0)-1 35i (ax-xx)*_2r (ar—ar)'; or APMKA=3 APNOA-4 ANP= 3 AONA – ANP. Consequently the infinitely extended space MKABQ is triple the generating semi-circle ANB.

Ex. VI. In the logarithmic curve yx=my, and fyr, or ABMP=my+C. But when y=1= AB, the space ABMP becomes

Therefore C=-m, and ABMP=m (7-1)= the rectangle OIQM. If we make y=0, we shall have the infinitely extended space BXYA=-m= the rectangle PQIT.

Ex. VII. Let thee be proposed the curve BM, whose equation is y=x*, we shall have (art. 84), the space ABMP = 1x*=

25
* (1 – +&c.)

44

M

zero.

B

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=1*+

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And when AP=PM=I, we have the space

1 1 ABMP=1

-&c.=0.7834 30497589.

33 Ex. VIII. In the curve

M

N of sines AMA'M, &c. of which the equation is y=sin x, we shall have

р APM= sinx=C-cosx.

= = If we make r =0, we

MI

N

n

ME

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votain C=1, and APM=1.-cosor. Let x=180° -7, and we shall have AMA'A=2=twice the square of the radius. If we suppose x=27=AA", we shall have the space AMA'B + A'M'A'A'=o, as is evident, since one is positive, and the other negative. In general, if x=2ka, the space will equal zero, and if x = (2k +1) 7, the space will =2.

If for the origin of 2 we take the

MB point A, the middle of A'A', weshall have

PA y = cos r. Consequently the space ABMP=sin x, the space ABA'A=1, and AMBA'A=0, or =2, if we leave out of consideration that of its two parts, one is positive, and one negative.

B 109. If the ordinates proceed from a fixed point C, we may find the quadrature of the curve as follows. Draw two radii CM, Cm, and from the centre c, and with the radius CM

А

2 describe the arc Mr; then the triangle CMm= Mr X CM + Mmr.

But 2 when the point m is indefinitely near to the point M, the space Mmr vanishes, and there remains flux

Mr X CM (COMC)= Let

N 2 then Mr=i, CM=y, and we shall have COMCE 3 Syž +C.

If we denote by Q the angle formed by CM with a fixed line issuing from the point C, or the arc which measures this angle in a circle whose radius is 1, we shall have Mr =yo, and COMC= Syy ©+C.

110. Ex. I. Let the curve AM be the conchoid, Pits pole, call PM =y, QM-a, PB=b, and the angle APM=. We shall 9

B 6 have 0:6::1: PQ=

P

M

; and

.

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6

62
Ła. Consequently the space APM=S
cos o

cosa
02
+

tan o + abl tan (45° + P) + to without any constant. Therefore + APMFPBQ, or ABQM=abl tan (46° +} $)+", and AAMM=2abl tan (45+10).

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Ex. II. In the cissoid, if we make AM = y, MAB = 0; AQ =

a

=

COS

O

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a’ sin?

COS? P

AO = MQ = a cos Q, and y = y

M

N -- cos p

Q, therefore
COS

a AKMOA=IS


-faa + $faa

A

P B cos' p = {aa tano caa + } aa (I sin o cos 0+40)= aa tan + aa sin 20-a y. Hence AKMPA =

Hapa sin 2 titzar sin 40; and the indefinitely extended space ŤKABQ= a p=3 AONB.

Ex. III. In the spiral of Archimedes, AGFBN=x AGFBA = 0, CM = y, CA = a, Mr = yi, flux

17

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