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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- ? nate mp, and the line Mr parallel to Pp; we 2 shall then have the surface of the space Mmp P MPxPp+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 become zero till the point m falls upon

A

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the point M; then M mp P will become the fluxion of the space AMP; Pp will be x, and we shall have flux (AMP)=yr, and consequently AMP=fyx+C, which by Art. 74

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Therefore also the space AQM = sxy=C+xy_Ya x + y3

--&c.

2y 2.3.y

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Therefore AQMP=ax-·

Make ro, and we shall have AQMP=0, and consequently Co.

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Ex. II. In the ellipse y = √ (aa-xx).

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-&c.)

Ex. III. In the parabola yx=p*x1x,

1 3

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and syx=f p3 x2 = xy, or the space APM is two-thirds of the circumscribed rectangle. The equation which comprehends 2 parabolas of all degrees is y=x" a"~"; therefore, mlynix + (m-n) la, conse

my quently=

y

nx

Α

m

PP

orm:n:: yx: xy :: fyx : fxy :: 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

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and yx=x1x (a—x)—1; consequently fyx, or the space AKMPA = x2 x (ax). Now fx (ax-xx) = the semi-segment AONP; and by art (80) we shall find that

ƒx*x (a—x)1=z x3 (a—x)*+ }ƒx2 x (a—x). Hence fr3 (a—x)— —

-

3 ·

=

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3ƒx (ax—xx)* —2x (ax—xx)*; or APMKA=3APNOA—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 fyx, or ABMP=my+C. But when y = 1 = AB, the space ABMP becomes zero. Therefore C=-m, and ABMP=m (y—1)= the rectangle OIQM. If we make y=o, we shall have the infinitely extended space BXYA=-m the rectangle PQIT.

Y

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AT

2

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xx

+ -&c.) 32 43

P

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otain C=1, and APM=1-cos r. Let x=180°, and we shall have AMA'A=2=twice the square of the radius. If we suppose x=2 π=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=2k, the space will equal zero, and if x = (2k+1) π, the space will =2.

of

If for the origin r we take the point A, the middle of A'A', we shall have

y= cos x. Conse

quently the space

M B

PA

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.

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Syx+C.

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If we denote by 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 Mry 9, and COMC= ¿ƒyy &+C.

110. Ex. I. Let the curve AM be the conchoid, Pits pole, call PM=y, QM=a, PB 6, and the angle APM. We shall

have b1; PQ =

b

; and

Cos

M

B

M

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without any constant. Therefore + APM PBQ, or ABQM=abl

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and AAMM=2abl tan (45° +9).

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