p=a, BC

a {(1+-1}, which is

From the nature of evolutes AB+BC=MC. Therefore BC=

MC-p= -4p. But MN=v(pr+pp) = ({p2+i pp).



9 Therefore calling 26


27 an expression for the length of any arc of the second cubical

parabola of which the equation is zd=au'.

36. Let the curve AMBа be a cycloid, and BODOʻits generating circle, let also MOP be an ordinate

M perpendicular to BD. If we make BP=x,

a PM=Y, BD= 2a, we shall have y= BO+V (2 ar-xx); now the fluxion of the




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arc BOisa

flur sin BO

cos BO




-fluxn (2 ax-xx

Therefore y=(2 a_r)

(2 ar- -IX)


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(2 ax-xx), the Auxional equation of the cycloid.

This premised, to find MC, the radius of curvature, suppose x constant, and taking the fluxion we shall have y =


XV (2 ax-xx) ji= 2 ait Consequently the radius MC--(**+y)}

*.XY 2 V2 a (22_x)=2 OD: Now MNC is parallel to OD, since (art 72 page 431) the tangent MT is parallel to OB; consequently OD-MN-NC.

Hence it follows 1st, that at the point A the radius of the evolute is zero, and consequently that the evolute passes through this point. 2ly. That the radius of the evolute at the point B is the line BE, double the line BD.

37. To determine the evolute ACE, complete the rectangle AE, and on the side AB'=DE=BD, as a diameter, describe a semicircle A'Q'B', draw AQ parallel to CM, and join C and Q'; this done, the angle NAQ'=NDO. Therefore OĎ=AQ, and the arc OID, or the right line AN-arc ALQ. Now OD=CN, therefore CN=AQ', and consequently CQ=AN=the arc ALQ'; a property


shewn (32)

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peculiar to the common cycloid. Hence the evolute ACE is a semicycloid equal to AMB, the one proposed, and differing from it only in position. We should have arrived at the same result, if we had directly investigated the equation of the evolute, by what was

The arc ac=MC=2 AQ'; therefore any arc of the cycloid is double the corresponding chord of the generating circle. Thus MB2 OB, AMB=2 BD, and the whole cycloid A Ba is quadruple the diameter BD.

38. Let the curve be thelogarithmic spiral ADM whose centre is A; we shall have

y cot MmA

and Mr

B taking the fluxion (supposing a constant) we shall have

y =0, and the radius of evolution MC

becomes ļ (2+ + ja). Therefore if we (2+y) - yry draw AC perpendicular to MA, and MC perpendicular to the tangent at M,their point of concourse C will be in the evolute; for the similar triangles Mrm and MAC give

Mm: Mr::MC : MA, that is ; or (32+ya): 0 ::MC: y

z2 Therefore MC=\n (i + %') as before.

39. The angle ACM=90°-AMC=AMT; from which it follows that the evolute ABC is the same logarithmic spiral ADM; it is merely disposed in a different manner. Hence the tangent MC is equal in length to the spiral ABC, though this last makes an infinite number of revolutions about the point A. Hence also, if we draw AT perpendicular to AM, we shall have MTarc ADM. Consequently the logarithmic spiral and the cycloid are their own evolutes.

y (x2+y+) {



40. If a curve from being convex, becomes concave, the point M at which this change occurs, is called a point of inflexion.

In order to determine this sort of points, we may conceive the tangent at M as being at once a tangent to the two parts MA, MO; and upon this supposition we may imagine on both sides of the point M two elements Mm, Mm in one right line; from which it follows that the radius of evolution or curvature at the point M must then be infinite. But as these elements

M may also be supposed to become gradually less and less, so as both ultimately to vanish, the radius of curvature must then be reduced to zero. 41. Therefore at the point of in

A P flexion the radius of curvature is always either infinite or nothing.

& .

Consequently supposing : constant, we shall always have (**+ya)

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Hence we must find the second fluxion of the equation of the curve, supposing ¿ constant, and we shall have a finite value of

-which must be put equal to either zero or infinity. By means of this equation and that of the curve, we shall be able to determine those values of x and y, which belong to the point or points of inflexion if there are several of them.

42. When the ordinates proceed from a fixed point, we shall have z? + y - y

- y 2 Ex. I. Let there be given the first cubical parabola whose equation is yo=a 'x, we shall have y=x} a, y=sx-ixas, and j=-31 aš, d, =-3x+ sy aa=0, at the point of

=, inflexion; therefore x=0. And consequently the point of inflexion is at the origin or vertex.

Ex. II. Let the curve be the conchoid of Nicomedes, of which the equation is y=6+27 (aaxx). First, taking the fuxion we



and y

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have y=

- (aab + x)

-; son (aa-3)


and again supposing i castant, we


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have_ _ _arø+32 °6 x?_?a 45

=0 at the point of inflexion. (a’zi_X")v (aa-xx) Consequently x? +36 x*—2a b=0, an equation which being resolved will give for x the value that belongs to the point of inflexion.

Ex. 1. Required the point of inflexion in the curve of which the

ara equation is y=

a' + x Ex. 2. Find the point of inflexion in the curve whose equation is ay ay (ax +.r')?

Ex. 3. Required the point of contrary flexure in the curve defined by the equation ay = a*x + x3?

43. If the ordinate MP of any curve BM

moin is greater or less than

K those ordinates which precede it, as pm, and than those which follow it, as p'm, it is B

D then called the maximum or minimum ordinate, that is, the


PPP greatest or least of the ordinates. thod which teaches us to determine these sorts of quantities is called the method of maxima and minima.

44. If CM is the radius of the osculating circle, or circle of equal curvature, it is evident that the ordinate MP must be greater or less than any other ordinate corresponding to any point of the arc KMD, described with the radius CM; cons

onsequently the ordinate MP (prolonged in the case of a minimum) passes through the centre of the osculating circle: therefore the tangent at M is parallel to the axis AP, and consequently the subtangent

Henee Y=0.


y Now y may be considered as any function of the abscissa AP (*); therefore to determine in what cases a quantity y depending upon I may become a marimum or minimum, we must find the Auxion of the equation which expresses the relation that they bear to each other, and make the quantity equal to zero. The resulting equation combined with the original one, will give the values of x and y, in which y is a maximum or minimum.

45. But to distinguish which case takes place, ne must observe


The me


that at a maximum point the radius of curvature is positive, and that it is negative at a minimum point. Now the expression for the oscu

(1+y) lating radius is

; and as
-0, we have CM-


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Therefore if y is a maximum,

is a maximum, y, must be negative, and if it is a minimum,must be positive.

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If it should happen that y is infinite or

X* nothing, M will then be a point of inflexion or of reflexion, and the tangent at M will be parallel to the axis; but in this case it may also occur that MP is neither a maximum nor minimum. See article 40.

46. It sometimes happens that the ordinate PM may be a maximum or minimim when the tangent at M is perpendicular to the axis. In this


B y

Y and consequently y=c; a formula which determines this sort of ordinates. Here MP may be considered both as a maximum and a minimum with respect to the two branches MB, MB. But this is a particular case included in the one of which we have spoken, and of which we shall proceed to give a few examples.

47. Ex. 1. Let it be required to divide a right line a into two parts such that their rectangle may be either a maximum or a minimum.

Call one of the parts x, the other will be amt, and we shall have ats for the expression of the maximum or minimum. Let then y=axts, and we shall have y=-2x=0, whence x=la. Now in order to determine whether this solution gives a maximum or minimum, take the fluxion of the equation y=-2x, and we have



=-2, a negative quantity ; from which it follows that the value

= gives a maximum y=fa.

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