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real or imaginary, and since in our case three of these are real and coincident, the circle of closest contact cuts the curve again in some one real fourth point. But it may happen as in the case in which the three ultimately coincident points are at an end of one of the axes of the conic that the fourth point is coincident with the other three, in which case the circle of closest contact has a contact of higher order than usual, viz., of the third order, cutting the curve in four ultimately coincident points, and therefore on the whole not crossing the curve. The student should draw for himself figures of the circle of closest contact at various points of a conic section, remembering from Geometrical Conics that the common chord of the circle and conic, and the tangent at the point of contact make equal angles with either axis. The conic which has the closest possible contact is said to osculate its curve at the point of contact and is called the osculating conic. Thus the circle of curvature is called the osculating circle, the parabola of closest contact is called the osculating parabola, and so on.

297. Analytical Conditions for Contact of a given order. We may treat this subject analytically as follows.

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be the equations of two curves which cut at the point P(x, y).

Consider the values of the respective ordinates at the points P1, P, whose common abscissa is x+h.

Let

Then

2

MN=h,

NP1=4(x+h),
NP2=√(x+h)

and

P2P1=NP1- NP2=4(x+h)− y(x+h)

1

1

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1

If the expression for P2P1 be equated to zero, the roots of the resulting equation for h will determine the points at which the curves cut.

If p(x)=√(x), the equation has one root zero and the curves cut at P.

If also p'(x)=√(x) for the same value of x, the equation has two roots zero and the curves cut in two contiguous points at P, and therefore have a common tangent. The contact is now of the first order.

If also "(x)="(x) for the same value of x, the equation for h has three roots zero and the curves cut in three ultimately coincident points at P. There are now two contiguous tangents common, and the contact is said to be of the second order; and so on. Similarly for curves given by their polar equations, if r=f(0), r=p(0) be the two equations, there will be n+1 equations to be satisfied for the same value of 0 in order

that for that value there may be contact of the nth order,

viz.,

ƒ(0) = p(0), ƒ'(0) = p′(0), ƒ"(0) = 4′′(0),

298. Osculating Circle.

fn(0)= p2(0).

The circle of curvature may now be investigated as the circle which has contact of the second order with a given curve at a given point.

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to be the equation of the curve.

Let

2
(x-x)2+(y-ÿ)2 = p2

be the equation of the circle of curvature.

By differentiating (2) we have

and differentiating again

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dy

(3)

x−x+(y−ÿ) 0,

d x

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dy day of equations (2), (3), (4) refer to

Now the x, y, a dx2

dx'

the circle. But, since there is to be contact of the second

order with the curve y = f(x) at the point (x, y),

have the same value as when deduced from

to the curve, i.e., we may write f'(x) for

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dy

d2y

and dx dx2

the equation dy and ƒ"(x)

dx

1+

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d2y

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dx2

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such a sign being given to the radical as will make

d2y

P

positive, i.e., if be positive we must choose the +

dx2

d2y

sign for the numerator, and if be negative we must

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dx2

The values of x and y are the same as those found geometrically in Art. 285, viz.,

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299. Conic having Third Order Contact at a given point. The locus of the centres of all conics having third order contact with a given curve at a given point (ie., cutting the curve in four ultimately coincident points) is a straight line which passes through the point of contact.

Let P be a point on the curve and C the centre of one of the conics having third order contact with the given

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curve at P. Let CD be the semiconjugate to CP and CY a perpendicular on the tangent at P.

Let CP=r, CD=r', CY=p, and let PC make an angle with the normal at P.

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dr

for

cos CPT

ds

- sin p, the arcs of the curve and of

the conic being measured from the points 0 and O' up to

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where de is found for one of the conics.

dp

ds

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