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Hence the area swept out by p during one revolution of the generating circle is

c (c’ – a) 7 Tb (Q2 + 3 ab + 2b).

4a a Subtracting from this the area of the sector of the fixed circle which is aab, we have for the area included between the epicycloid and the fixed circle

A = * . * (sa + 26). When a curve forms a loop, the area may sometimes be conveniently found by taking , or the tangent of the angle which the radius makes with the axis of v, as the independent variable. If we put = tan 0 = t, we have do = dt cos° and

A = } [rodo = } SdOxo sec° 0 = } fatxo. (21) The curve yê – 3axy + x3 = 0, has a loop which touches the axes of w and y at the origin ; see fig. 51. Now putting y = xt, we find




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ai 6t

A = 1 sdtx - 1 6

The whole

and for each loop the limits of t are mand area is ab + (a1o) tan

Sect. 2. Rectification of Curves. When a curve is referred to rectangular co-ordinates, the length of any portion of it is found by integrating

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between the proper limits.
(1) The equation to the common parabola being

yo = 4mx,
the length of an arc measured from the vertex is
, 1x + m 2

m, m + 2 {x + (x? + mx)!} -) = (22 + m x) + = log

(2) The equation to the semicubical parabola is

ay' = x3.
The length of the arc measured from the origin is

_(10 + 9x)? – (4a)
s =

27a: This was the first curve which was rectified. The author was William Neil, who was led to the discovery by a remark of Wallis in his Arithmetica Infinitorum. See Wallisii Opera, Tom. I. p. 551. (3) To find when the curves expressed by the equation

"Y" = m+" are rectifiable. We have

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the second gives

m +n
***= { = f = = &c.

(4) The equation to the cycloid being

dy (2ax – v?)

doo we have

$ = 2 (20x)+ C. Hence the whole length of the cycloid is 8a or four times the diameter of the generating circle. This rectification was discovered by Wren. (5) The equation to one of the hypocycloids is

wi + y} = a); the whole length of the curve is 6a.

(6) The equation to the catenary is


and 8 = (y2 - 0 the arc being measured from the point where y = C.

Hence as the area is equal to c(y – c)!, it is equal to 08; that is, the area contained between the axes, the curve and any ordinate is equal to the length of the corresponding arc multiplied by a constant.

(7) The equation to the tractory is

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supposing the arc to be measured from the point where y = a. When a curve is the evolute of another curve, the length of its arc is best found by taking the difference of the radii of curvature of the involute, corresponding to the extremities of the arc.


To find the length of the evolute of the ellipse.

The radius of curvature of the ellipse at the extremity of the major axis is ; that at the extremity of the minor axis is

: therefore the length of the fourth part of the evolute is a 62 a13 ū-arab. If the curve be referred to polar co-ordinates r and 0,

drin S = (do

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supposing it to be measured from the pole. Hence the arc is equal to the portion of the tangent at its extremity, which is intercepted between the point of contact and the subtangent.

(10) The equation to the spiral of Archimedes being

r = ad, the length of the are from the origin is »» (c' + roo) a, p + (a + po?)!


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This is the same as the arc of a parabola (whose latus rectum is 2a) intercepted between the vertex and an ordinate equal to r. (11) In the involute of the circle the equation to which is

go? – p= a,

If the arc be measured from the point where r = a we find c= -, and 8 = Now p is always equal to the length of the arc of the circle which is unwound, so that if this be called ad,

8 = 1a0%.
(12) The equation to the epicycloid is

c(po – a)

c - a ' where c = a + 2b, a and 6 being the radii of the fixed and generating circles respectively. Hence

(c' - a?) rdr -_(c“ – a*) (C – po?) . 8= a J (cu — q)] ** a

The whole arc corresponding to one revolution of the generating circle is 8?- (a + b).

The corresponding arc in the hypocycloid is 8(a - b).
For a curve of double curvature we have


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