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If the axes be oblique, it is seen in like manner (Art. 158) that the condition for similarity is

ab - h2

= 2

a'b' - h'

(a+b- 2h cos w)" (a' + b' — 2h' cos w)*

It will be seen (Arts. 74, 154) that the condition found expresses that the angle between the (real or imaginary) asymptotes of the one curve is equal to that between those of the other.

THE CONTACT OF CONIC SECTIONS.

238. Two curves of the mth and nth degrees respectively intersect in mn points.

th

For, if we eliminate either x or y between the equations, the resulting equation in the remaining variable will in general be of the mnt degree (Higher Algebra, Art. 73). If it should happen that the resulting equation should appear to fall below the mnth degree, in consequence of the coefficients of one or more of the highest powers vanishing, the curves would still be considered to intersect in mn points, one or more of these points being at infinity (see Art. 135). If account be thus taken of infinitely distant as well as of imaginary points, it may be asserted that the two curves always intersect in mn points. In particular two conics always intersect in four points. In the next Chapter some of the cases will be noticed where points of intersection of two conics are infinitely distant; at present we are about to consider the cases where two or more of them coincide.

Since four points may be connected by six lines, viz. 12, 34; 13, 24; 14, 23; two conics have three pairs of chords of intersection.

239. When two of the points of intersection coincide, the conics touch each other, and the line joining the coincident points is the common tangent. The conics will in this case meet in two real or imaginary points L, M distinct from the point of contact. This is called a contact of the first order. The contact is said to be of the second order when three of the points of intersection

G G.

coincide, as, for instance, if the point M move up until it coincide

T

T

T

M

with T. Curves which have contact of an order higher than the first are also said to osculate; and it appears that conics which osculate must intersect in one other point. Contact of the third order is when two curves have four consecutive points common; and since two conics cannot have more than four points common, this is the highest order of contact they can have.

Thus, for example, the equations of two conics, both passing through the origin and having the line a for a common tangent are (Art. 144)

ax2 + 2hxy + by2+2gx=0, a'x2+2h'xy + b'y2 + 2g'x = 0. And, as in Ex. 2, p. 175,

x {(ab' — a'b) x + 2 (hb' − h'b) y + 2 (gb' — g′b)} = 0,

represents a figure passing through their four points of intersection. The first factor represents the tangent which passes through the two coincident points of intersection, and the second factor denotes the line LM passing through the other two points. If now gb'g'b, LM passes through the origin, and the conics have contact of the second order. If in addition hb'h'b, the equation of LM reduces to x = 0; LM coincides with the tangent, and the conics have contact of the third order. In this last case, if we make by multiplication the coefficients of y" the same in both the equations, the coefficients of xy and x will also be the same, and the equations of the two conics may be reduced to the form

ax2 + 2hxy + by* + 2gx=0, a'x2+2hxy + by2 + 2gx=0.

240. Two conics may have double contact if the points of intersection 1, 2 coincide and also the points 3, 4. The condition that the pair of conics considered in the last Article should touch at a second point is found by expressing the condition that the line LM, whose equation is there given, should touch

either conic. Or, more simply, as follows: Multiply the equations by g' and g respectively, and subtract, and we get

(ag' — a'g) x2 + 2 (hg' — h'g) xy + (bg' — b'g) y3 = 0,

which denotes the pair of lines joining the origin to the two points in which LM meets the conics. And these lines will coincide if

(ag' — a'g) (bg' — b'g) = (hg' — h'g)3.

241. Since a conic can be found to satisfy any five conditions (Art. 133), a conic can be found to touch a given conic at a given point, and satisfy any three other conditions. If it have contact of the second order at the given point, it can be made to satisfy two other conditions; and if it have contact of the third order, it can be made to satisfy one other condition. Thus we can determine a parabola having contact of the third order at the origin with

ax2+2hxy + by2 + 2gx = 0.

Referring to the last two equations (Art. 239), we see that it is only necessary to write a' instead of a, where a' is determined by the equation a'b = h2.

We cannot, in general, describe a circle to have contact of the third order with a given conic, because two conditions must be fulfilled in order that an equation should represent a circle; or, in other words, we cannot describe a circle through four consecutive points on a conic, since three points are sufficient to determine a circle. We can, however, easily find the equation of the circle passing through three consecutive points on the curve. This circle is called the osculating circle, or the circle of

curvature.

The equation of the conic to oblique or rectangular axes being, as before,

ax2+2hxy+by+2gx=0,

that of any circle touching it at the origin is (Art. 84, Ex. 3)

x2 + 2xy cos w + y2 — 2rx sin w = 0.

Applying the condition gb'g'b (Art. 239), we see that the

condition that the circle should osculate is

-g

g=-rb sin w, or r= b sin w

The quantity r is called the radius of curvature of the conic at the point T.

242. To find the radius of curvature at any point on a central

conic.

In order to apply the formula of the last Article the tangent at the point must be made the axis of y. Now the equation referred to a diameter through the point and its conjugate

(+3=1) is transferred to parallel axes through the given

a

point, by substituting x + a' for x, and becomes

x+a'

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Therefore, by the last Article, the radius of curvature is

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a' sin w

Now a' sino is the perpendicular from the centre on

the tangent, therefore the radius of curvature

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243. Let N denote the length of the normal PN, and let y denote the angle FPN between the normal

and focal radius vector, then the radius of

Q

curvature is

N cos**

bb
For N: (Art. 181),

F

=

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α

and cos (Art. 188), whence the truth of the formula is manifest.

*In the Examples which follow we find the absolute magnitude of the radius of curvature, without regard to sign. The sign, as usual, indicates the direction in which the radius is measured. For it indicates whether the given curve is osculated by a circle whose equation is of the form

x2 + 2xy cos w + y2 2rx sin w = 0,

the upper sign signifying one whose centre is in the positive direction of the axis of x; and the lower, one whose centre is in the negative direction. The formula in the text then gives a positive radius of curvature when the concavity of the curve is turned in the positive direction of the axis of x, and a negative radius when it is turned in the opposite direction.

Thus we have the following construction: Erect a perpendicular to the normal at the point where it meets the axis; and again at the point Q, where this perpendicular meets the focal radius, draw CQ perpendicular to it, then C will be the centre of curvature, and CP the radius of curvature.

244. Another useful construction is founded on the principle that if a circle intersect a conic, its chords of intersection will make equal angles with the axis. For the rectangles under the segments of the chords are equal (Euc. III. 35), and therefore the parallel diameters of the conic are equal (Art. 149), and therefore make equal angles with the axis (Art. 162).

Now, in the case of the circle of curvature, the tangent at T (see figure, p. 226) is one chord of intersection and the line TL the other; we have, therefore, only to draw TL, making the same angle with the axis as the tangent, and we have the point L; then the circle described through the points T, L, and, touching the conic at T, is the circle of curvature.

This construction shows that the osculating circle at either vertex has a contact of the third degree.

Ex. 1. Using the notation of the eccentric angle, find the condition that four points a, ß, y, d should lie on the same circle (Joachimsthal, Crelle, XXXVI. 95).

The chord joining two of them must make the same angle with one side of the axis as the chord joining the other two does with the other; and the chords being

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we have tan (a + ß) + tan§ (y + d) = 0 ; a + ß + y + d = 0 ; or = 2mπ.

Ex. 2. Find the coordinates of the point where the osculating circle meets the conic again.

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4x'3 By; hence 8=-3a; or X= a2

· 3x'; Y=

4y's
b2 -3y'.

Ex. 3. If the normals at three points a, ß, y meet in a point, the foot of the fourth normal from that point is given by the equation a + B + y + d = (2m + 1) π.

Ex. 4. Find the equation of the chord of curvature TL.

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Ex. 5. There are three points on a conic whose osculating circles pass through a given point on the curve; these lie on a circle passing through the point, and form a triangle of which the centre of the curve is the intersection of bisectors of sides (Steiner, Crelle, XXXII. 300; Joachimsthal, Crelle, XXXVI. 95).

Here we are given d, the point where the circle meets the curve again, and from the last Example the point of contact is ad. But since the sine and cosine

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