and, in like manner, that the square of a parallel semi-diameter of the second is equal to another constant divided by

a' cos 0+2h' cos sin + b' sin20.

The ratio of the two cannot be independent of 0 unless

Hence two conic sections will be similar and similarly placed, if the coefficients of the highest powers of the variables are the same in both or only differ by a constant multiplier.

235. It is evident that the directions of the axes of these conics must be the same, since the greatest and least diameters of one must be parallel to the greatest and least diameters of the other. If the diameter of one become infinite, so must also the parallel diameter of the other, that is to say, the asymptotes of similar and similarly placed hyperbolas are parallel. The same thing follows from the result of the last Article, since (Art. 154) the directions of the asymptotes are wholly determined by the highest terms of the equation.

be

Similar conics have the same eccentricity; for m2 a2 - m2b2

=

m*a*

Similar and similarly placed conic sections

have hence sometimes been defined as those whose axes are parallel, and which have the same eccentricity.

If two hyperbolas have parallel asymptotes they are similar, for their axes must be parallel, since they bisect the angles between the asymptotes (Art. 155), and the eccentricity wholly depends on the angle between the asymptotes (Art. 167).

236. Since the eccentricity of every parabola is =1, we should be led to infer that all parabolas are similar and similarly placed, the direction of whose axes is the same. In fact, the equation of one parabola, referred to its vertex, being y2=px, or P cos e sin'0'

P =

it is plain that a parallel radius vector through the vertex of the other will be to this radius in the constant ratio p' : p.

Ex. 1. If on any radius vector to a conic section through a fixed point O, OQ be taken in a constant ratio to OP, find the locus of Q. We have only to substitute mp for p in the polar equation, and the locus is found to be a conic similar to the given conic, and similarly placed.

The point may be called the centre of similitude of the two conics; and it is obviously (see also Art. 115) the point where common tangents to the two conics intersect, since when the radii vectores OP, OP' to the first conic become equal, so must also OQ, OQ' the radii vectores to the other.

Ex. 2. If a pair of radii be drawn through a centre of similitude of two similar conics, the chords joining their extremities will be either parallel, or will meet on the chord of intersection of the conics.

This is proved precisely as in Art. 116.

Ex. 3. Given three conics, similar and similarly placed, their six centres of similitude will lie three by three on right lines (see figure, page 108).

Ex. 4. If any line cut two similar and concentric conics, its parts intercepted between the conics will be equal.

Any chord of the outer conic which touches the interior will be bisected at the point of contact.

These are proved in the same manner as the theorems at page 191, which are but particular cases of them; for the asymptotes of any hyperbola may be considered as a conic section similar to it, since the highest terms in the equation of the asymptotes are the same as in the equation of the curve.

Ex. 5. If a tangent drawn at any point P of the inner of two concentric and similar ellipses meet the outer in the points T and T', then any chord of the inner drawn through P is half the algebraic sum of the parallel chords of the outer through T and T'.

237. Two figures will be similar, although not similarly placed, if the proportional radii make a constant angle with each other, instead of being parallel; so that if we could imagine one of the figures turned round through the given angle, they would be then both similar and similarly placed.

To find the condition that two conic sections, given by the general equations, should be similar, even though not similarly placed.

We have only to transform the first equation to axes making any angle with the given axes, and examine whether any value can be assigned to which will make the new a, h, b proportional to a', h', b'. Suppose that they become ma', mh', mb'.

Now, the axes being supposed rectangular, we have seen (Art. 157) that the quantities a+b, ab - h2, are unaltered by transformation of coordinates; hence we have

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

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 mn 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 x 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+by2+2gx=0, a2x2+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+by+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= h3.

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 + by2 + 2gx = 0,

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

x2 + 2xy cos w + y2 - 2rx sin ∞ = 0.

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